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NUMERICAL EVALUATION OF I S F I N I T E SERIES AND ISTEGRALS K H I C H ARISE I N CERTAIS PROBLEMS O F LISEAR HEAT FLOW, ELECTROCHEMICAL DIFFUSIOS, ETC. BY W. LASH MILLER AND A. R . GORDON

I n the Introduction to his Thdorie analytique de la chaleur’ Fourier points out that unless the solutions of the problems he considers are presented in a form that permits of numerical computation, they will be-for many purposes-valueless. He says :“Interprktation numkrique des r6sultats du calcul est nkcessaire et c’est un degrk de perfection qu’il serait tr&s important de donner A toutes les applications de 1’Analyse au Sciences naturelles. On peut dire que, tant qu’on ne l’a pas obtenu, les solutions demeurent’ incompletes ou inutiles, et que la d r i t k qu’on se proposait de decouvrir n’est pas moins cachee dans les formules d’Analyse quelle ne 1’6tait dans la question physique elle-m&me”* Fourier’s own contribution consisted in finding functions which satisfied the differential equations for the flow of heat and certain “boundary” and “initial” conditions as well. But in claiming? that these functions “are -+given in a form that is [always] convenient for numerical computation” he overstates his achievement; most of his solutions involve infinite series which except for special values of the const,ants converge too slowly to be summed. Subsequent writers seem to have lost sight altogether of Fourier’s criterion; for instance the function given by Byerly3 as the solution of the problem discussed in his Art. j~ e x . 4 involves integrals which are not only untahulated but which in general are not easily evaluated by the ordinary processes of graphical or tabular integrat’ion. The solutions obtained by Fourier’s method for certain electrochemical problems on which we have been working in this laboratory, shared both defects; and in searching for methods by which the functions contained in them could be evaluated with the least expenditure of labour, we were led to study the whole question from a general point of view insofar as problems of linear flow are concerned. Fourier’s solutions, and the solutions obtained by his method for certain electrochemical problem^,^ are in general presented as infinite series; in Part I1 J

‘Croxned hv the Academy in 1812, printed in 1824-26. First published hp Fermin Didot, Paris ( I $ z P ) . English edition: “The analytical theory of heat” (18781; See also Oeuvres de Fourier, Vol. I (1888), Vol. I1 (1890). Oeuvres de Fourier, Vol. I, p. 9. W. E. Byerly: Elementary Treatise on Fourier’s series etc. (1893). 4,T.R. Rosehrugh and W. Lash Miller: Mathematical theory of the changes of concentration at the electrode brought about by diffusion and by chemical action, J. Phys. Chem., 14, 816-884 (1912).

2786

W. LASH MILLER AND A .

R. GORDON

of the present paper we examinel the mathematical expressions for these series in the special case that the flow is linear, and shew that they fall into, or are closely related to, two groups for which we propose the names “Series of Group” and “Series of the A E Group” respectively. In Part I the initial and bound’ary conditions that give rise to series of these two groups are exemplified; other conditions? leading to solutions which do not involve series of these t,wo groups are not considered. The examination brings out the relationship which the series of the @ E and LIEGroups bear to Jacobi’s thetafunctions ; a relationship recognized by Jacobi3 and formulated by Poinear64 and by Carslawb! but hitherto not exploited as a source of methods for evaluating the Groups in question. These methqds (Parts I11 and IT)are founded on the existence of pairs of series whose sums are equal but whose convergences are very differently affected by changes in the values of the constants they contain. The existence of such pairs was discovered by Jacobi in the case of his thets-functions; it is not however necessary to follow his argument in detail; for, as shewn in See. 5 , Jacobi’s classical results are only a special case of a formula due to Poisson6 (“the Euler-Maclaurin Sum Formula with Poisson’s Remainder”) which in turn we shew to be merely a formulation of the Fourier’s cosineseries for a series.’ The systematic plan based on this foundation for approximating to the sum of any series of the Groups here considered, must perforce take into account certain practical difficulties; one of these is the lack of numerical tables of certain integrals, another is the lack of finite expressions for certain ‘ T h e examination is facilitated by the fact, noted hy Fourier in a memoir of 1829 [MBmoire s.1. theorie d.l.chaleur: M6m. de 1’Acad. roy.des Sciences 8, 581 (1829); Oeuvres de Fourier, Vol. I1 p . 1491 that his general solution can he expressed as the sum of three quantities each of which (taken hy itself) is the solution of a special case. This general solution, obviously an afterthought on the part of Fourier, 1s ignored by Byerly; probably because the 3fdmoire has heen omitted from editions of the Thdorze. 2 See H. S. Carslaw: Introduction to the mathematical theory of the conduction of heat in solids, art. 36 p. 74 and art. 105 p. 2 1 3 , (1921). 3 “[Ces formulas] ne seront pas sans inter& pour les cC.l&bresgkomittres qui s’occupent du mouvement de la chaleur; les numbrateurs et les dhominateurs des fonctions par lesquelles on a exprimi: les fonctions trigonombtriques de l’amplitude Btant souvent rencontres dans ladite question.” Letter from Jacobi to Legendre, April 1828 G. L. Jacobi’s gesammelte Werke, (r881), Vol; I p. 41jl. The numerators and denominators there referred t o are Jacobi’s theta-functions. H. Poincarb: Theorie analytique de la propagation de la chaleur, p . 91, (1895). Carslaw: loc. cit., pp. 160, 161. Mt.m.s.l.calcu1 numkrique des integrales dCfinies, MQm.d.1’Acad.roy.des Sciences, 6, 5 7 1 (18231. Poisson harely escaped anticipation bv Legendre [see .4. M. LeGendre: Exercices de cslcul intbgral, Vol. 11, Paris (1817) p. ;jo Eq. g]. His deduction was hnsed upon a theorem which has all but disappenred from mathematical literature, and whose validity has been attacked [see Carslaw: Introduction to the theory of Fourier’s series and integrals, 2nd Ed., (1921), sec. 99, p. 2301. Boole deduced the formula by a method which the editor of the third edition [G. Boole: Treatise on the calculus of finite differences, 3rd Ed., sec. 16, p. 154 (1880)] rejected; on insufficient grounds it seems to us. The deduction given helow involves but one assumption, viz. that the summand of the series to,he transformed can be expanded in a cosine series, and thus supplies an answer to the question often raised “With what restrictions is the Poisson formula to be employed?”. (Pee foot note page 2 787.)

[c.,

’

NL’MERICAL EVALUATION O F INFISITE SERIES

2787

simpler series here called “Stationary”. Where they exist, these obstacles restrict the choice of an approximation. Among the methods still available, the choice is based upon an a priori estimate of the amount of labour each procedure would require in order to obtain the sum with the degree of accuracy considered necessary by the computer; the fact that such an estimate can readily be made, renders the whole scheme practical.

Synopsis P a r t Z of the present paper gives examples of the problems1 which can be solved in terms of series of the Groups whose summation is discussed below, and tabulates the “Original” (i.e. Fourier’s) Forms of the solutions. Part ZI classifies the series, defines the Families and Groups of which they are members, and explains the symbols used to represent them. I t then deduces Poisson’s Formula (assuming the validity of Fourier’s expansion in a cosine-series) and uses it to obtain the “Poisson Forms’‘ of all the series of the Families considered, including the series which occur in the Solutions of Part I. Parts IZI and I T ‘ give labour-saving devices for evaluating all the aeries of the @ E and A E Groups, and those related serieswhich occur in the Solutions. Part T’ uses the methods developed in Parts I11 and IT’ to evaluate certain integrals. The ilppendix discusses the integral Jez’.dz where z is complex. I t contains numerical tables of F(z) = e-“.Jo”e”.dy, Table X S V I I I ; G(z) = l z e - a ’”.( 1/2/i).dy and J G ( x ) . d x , Table X S V I I ; and tables from which fourplace values of the Fresnel Integrals may be obtained for all values of the argument without recourse to graphical interpolation (Table XXIS-). Readers whose sole wish is to find, as quickly as possible, t,he most suitable approximation to the series included in a Solution of Table I, must consult Table IT- to find t’he symbol for the series; if this is not of the [eren] or [odd] “ere Thomson was the pioneer [Quar. Jour. Math., 1, 316 (1857)j. In 1 8 j 7 , without recognizing the functions with which he was dealing, he shewed that the “Original Form” of Jacobi’s 8 : can he deduced from the “Poisson Form” (see below p. 2 8 0 8 ) by developing the latter in a series of cosines. In a foot-note to this paper, Cayley pointed out that the relation found bv Thomson had alreadv heen ohtained by Jacohi and shewed how it might lie deduced fromuknown properties of the Elliptic Functidns. From the point of view of the present paper, the chief service here rendered by Cayley was in reminding workers in this field that their functions are related to the Thetas; he appiirentlv did not realize that Thomson’s method might be made the starting point for a simplified treatment of the Elliptics themselves. Carslaw [Conduclion of Heat, loc. cit., p. I 591 following essentially Thomson’s procedure, shewed that when f ( y ) is an “even” or “square” function

..~. Botil Carslaw’s and Thomson’s equations are special cases of Poisson’s Formula, see Table VI1 p. 2806. ’ Thomson shews that the motion of a viscous fluid, closed electric current within a homogeneous conductor, and the electric potential in the conductor of a submarine cable when electromagnetic inertia can be neglected, can he dealt with similarly [B. A. Report, Sept. 1888; Math. and Phys. Papers Vol. 111, p. 4281.

2788

W. LASH MILLER AND A . R. GORDON

type it must be replaced as shewn in the "Summary" of p. 2801. The form of j(u)in the expression for Y a t the head of Table I must then be noted, and the Class of the series found from Table X. The best approximation in view of the numerical values of at and g s is given in Table X. Sec. g furnishes examples of the use of each approximation, and a way to find an upper limit to the error involved in each. If the series to be evaluated is contained in one of the Solutions of Table 11, its symbol must be found from Table IV, and a suitable approximation from Part IV. Readers who require approximations to an integral of the form Jf(z2).dz or Jek"'f(z).dz, should consult Part V ; the approximations there given, however, are not in every case available, for they may involve integrals which themselves have not been tabulated. Series of the @.p and AF Groups do not occur in the Solutions of Part I ; but because of their resemblance in form to those which do occur, they have been treated briefly, in brackets. PART I THE PROBLEMS AND THE ORIGINAL FORMS OF THE SOLUTIONS The problems of heat flow and of diffusion which give rise to the series whose summation is discussed below may be grouped under four heads, the first second and third of which are solved by means of the differential equation ( I ) and the fourth by means of the differential equiition ( 2 ) . Sec. 1: Problems leading to the differential equation (I).

( I ) Problems that have to do with the ,jiozu o,f hzat through a straight rod of uniform material and cross-section with the proviso that no heat can pass through the surface except a t the ends:--in these the initial distrihution of temperature along the axis is specified, and i t is asmmed that the two ends of the rod are maintained a t arbitrary (Constant or varied) temperatures throughout the experiment, iir that they are impervious to heat, or that one end is impervious to heat while the temperature of the other is arbitrarily varied. Choosing as z-axis the direction of the rod (x:=o at one of its ends and z=1 a t the other) rising the letter z to denote the tempernture and t to denote the time, the Initial Condition can be put in the form: [at t = o ] z = F ( z ) ; and the Boundary Condition at one end, e.g. [at x = o ] is z = f ( t ) if the temperature a t the left hand end is arbitrarily controlled, or = o if that end is impervious to heat, while iat x = 11 either z = $ ( t ) or & / a x = o. i z j I'roblmw oj s i m p l e diflmion through a solution contained in a cylindrical vessel:-Here the axis of the cylinder is taken as the x-axis, of some selected constituent, supposed and z indicates the concent uniform throughout any giT;en cross-section of the vessel. The initial distribution of concentration along tile axis or" the cylinder is specified, so that the Initial Condition is [at != 01 z - P ( x ) as in the heat prot:!em. The ends of the cylindrical mass of solution are taken t o he imperyious to the constitucnt

NUMERICAL EVALUATION OF INFINITE SERIES

2789

whose concentration is denot,ed by z-either because they are bounded by the material of which the vessel is composed, or (if one of them is a free surface) because t,he constituent in question is assumed to be non-volatile; so that the Boundary Condition both [at 5 = 01 and [at x = 1 ] is d z j d x = 0 . ( 3 ) Problems of diflusion during electrolysis when the ends of the cylindrical vessel are bounded by electrodks; with the proviso that no chemical reaction requiring appreciable time for its completion occurs between the products of electrolysis and the electrolyte:-Here z is used to indicate the concentratTon of some constituent (assumed uniform throughout each crosssection] which is formed or removed at either or both electrodes by electrolysis or migration or both combined; so that the rate of formation (or removal) of this constituent is proportional to the current. The Boundary Condition a t an electrode a t TThich the constituent appears or disappears will be &Id.,:= f ( t ) ; at one where it is neither formed nor destroyed and to or from which it is not transported by the current, the Condition will be d z l d x = o . If the experimental conditions are such that during the electrolysis the concentration of the constituent to which z refers remains unaltered at some point between the electrodes, that point may be taken as the origin or as I = 1 ivhichever is preferred; the Boundary Condition there will be z = constant, and the conrentration changes between this point and either of the electrodes will obviously not, be affected by any changes that rnay occur between it and the other. The same conditions hold in the case of a cylindrical electrode rotating in a uniform field; for the electrode rnay be regarded as surrounded by an sclherent film of solution, uniform in thicknt+s, through which diffusion occurs: and under ordinary experimental conditions this film is so thin that the difference between the areas of its inner and outer surfaces mey be neglected. S r ~ l ~ t i oofn (~I ) , ( 2 ) and (3):-In thew three classes of problems the only Init,ial Condition that' occurs is z = F ( r i ; and the only Roundary Conditions are z = fit) and 13z;d.z = f ( t ) , if the expressions F i r ) and f(l) be taken to inc1i:de zero and constants.1 In solT-ing them it is assumed that

_The _ forms of ,Ll, nssiimed in the conventional heat problems are: f s I ) = ronstnnt, s m 6 - t t u , +, t 4 ' , s i n ; [ > ; - thr last tn'o in problems invulviiig radiation; the same four forms occur in e!ectrocheniiclil problems. If whrn t =1?, the temperature at which one end of the rod has heen maintained in suddenly changed (see Byerly: loc. rit., See. g r , Ex. 2 ) or the current IS abruptly varied, f:tj is discontinuous (see p. 2 8 3 3 ) , Fir' is discontinuous in diffusion experiments where a laver of water is superposed on one of solution. If a change is made in the nature of the Boundarv Conditions, i.e. if for t > 9, z is specified y. hile for t '.lkl

ditto

where B = Rem :[evenllc, X = gx - gX;

D = Rem:[odd]Ac, X=gx-gX;

+ ,-(z+X)*

4kI

} .F(X).dX - (D+E ) / 1

+ (B+C)/I C = Rem:[even]Ac,X=gx+gX E = Rem:[odd]Ac,X=gx+gX

NUMERICAL EVALUATION O F INFINITE SERIES

2813

The Poisson Forms of Table L Nos. Ia, za, 3a and 4a may be obtained from those of Table I Nos. I, 2, 3 and 4 respectively by replacing x in the latter by ( l - x ) ; this includes replacing a / a x by a / a ( l - x ) = - a / a x . PART I11 THE EVALUATION OF SERIES OF THE @E GROUP AND RELATED SERIES, INCLUDING THOSE OF TABLE I In Part 11, Sec. 7 it has been shewn how any series contained in the Solutions of Table I (or more generally any series of the @ E Group) can be replaced by another infinite series which we call the Poisson Form of the series in question; so that in evaluating the series, either the Original Form or the Poisson Form may be chosen as may prove convenient. I n practice this choice is nearly always reduced to one between the Original Form and the Pbapproximation, and is determined by the necessity of avoiding difficulties of summation or of integration, which in turn depend on the value of at and of gx and on the form of f(u);the present Part is devoted to a discussion of these matters in detail. If the value of at is small, the Original Form is inconvenient because of the slow convergence of the Evanescent Series (see p. 2803), the infinite series in the Poisson Form on the other hand is rapidly convergent and if the integrals involved have been tabulated this Form is immediately available (approx.i); in general, however, these integrals have not been tabulated; approz.ii, iii, iv and i x are in effect methods for evaluating them. If on the other hand the value of at is large, the slow convergence of the Poisson Form and the consequent inaccuracy of the P-approximation is in itself a sufficient reason for avoiding it; the Evanescent Series V : @ E on the other hand is rapidly convergent, and if the integral involved in V : @ E (the Original Integral, p. 2804) has been tabulated, and a finite expression for the Stationary Series S : @ Eis known, the employment of the Original Form is indicated (approx. v). In certain cases there is no finite expression for the sum of S : @ E , and for a few forms of j(u)the integral involved in V :@ E is untabulated; approx.vi-ix shew how these cases may be handled. In Sec. 8 the influence of the numerical values of at and gx on the magnitude of rem:@^ is discussed; this leads to a convenient method for finding the values of at and gx with which the P-approx. can be used without introducing errors greater than a quantity fixed in advance by the computer. In this Section also, relations are established which permit the evaluation of P: @ E in many cases where the integrals contained in it have not been tabulated. In Table X the series are classified according to the form of f(u); and this table lists for each Class the approximations most suitable for given values of at and gz. I n Sec. 9 the various approximations to @ E and to a / a ( g x ) . @ E are considered in detail and illustrated by numerical examples';

* These are intended t o illustrate the accurac of the mathematical methods, and are taken to a larger number of significant figures tzan is ordinarily necessary in laboratory computations. I n many caaes alternative methods of computation are used; these shew the great saving of labour that may be effected by following the methods recommended.

2814

W. LASH MILLER AND A. R. GORDON

the evaluation of S: is dealt with in Sec. I O ; Rules for finding upper limits to the Remainders neglected in these approximations are given in Sec. I I .

TABLE X Classification of Series of the Group according to the form of f(u);with the approximations recommended for each Class Class Definition Approximation recommended for values of 4; and gz A

A,

+ + . + . . . + .

1'

S

+ + + +

+ + + .

above BE when gz

. + .

. + . . .

.

.

.

2'

v v oii Vii ix ix

*o

below BE

above B below B when gz = o

i V OB J,ii;O JH,iii;OHE,iv v OB J , i i ; O J E , i i i V ix Vi ix vii ... ix ttu is ix

i i at

i is i ix

Examples

A. B.

C. D.

up.

E. __

, sin or cos b2ufa. eKu.(sin or cos) b2U+CU. UP.efb?u.

(b?+~)-~;

eb'U2

F. I v'( b2 +u)(c2 +u ). G. 1 / ( b 2 + u 2 ) .

( b * j = ~ ) * ~ + F(&&l). ~.~;

Definitions of the Classes:-The Class to which a given series belongs depends upon the form off(u), and is based upon the existence or nonexistence of numerical tables from which the Ap roximation Integral (p, 2811) the Approximation Integral when gz = o or the OriginarIntegral (p. 2804) can readily be computed, and upon the existence or nonexistence of a finite expression for the sum of the Stationary Series (p. 2803). The existence of such tables or of such an expression is indicated by a plus sign in the column headed A A , I' or S respectively. their nonexistence by a dot. The approximation integrals for a number of forms of f(u$ are tabulated in Table XI, Original Integrals in Tables XI1 and XIII, finite expressions for S in Table XII. The approximations are designated by the numbers i-iz (see Sec. 9). Table XVI shews that for each Class (except the last) more than one approximation is available; those given in Table X are recommended because they involve the least labour. The range of values of -/;and gz for which an approximation is recommended is indicated by the position of the point representing the series in Fig. I1 (in which the ordinate of E is to he chosen with r e g a r i t o the accuracy desired, see Table XIV). Thus "OBJ, ii" means that if the values of v'at and gz are those represented by any point within the triangle OBJ of Fig. 11, approz. ii is recommended for series of Class B and Class C when gz is not zero. The Ezamples include ail those dealt with in Tables XI, XI1 and XIII, and are all that are met with in the literature (see foot-note p. 2789); the letter p denotes a positive integer or zero, b2 c2 and K are real positive quantities. The Class to which any form of f(u) is assigned may obviously be altered by the publication of new tables; the appearance of numerical tables of H and K (see Appendix), for instance, would transfer up.e-b2u from Class B to Class A, and e K u . s i n or cos 6'u from Class C to Class B. Approximations to * ' E (seep. 2834) may also be chosen from this tahle; for this purpose i-iz must he replaced by ia-iza, and when approz. iv is recommended in the table, approz. iiia must be employed.


2815

For numerical values of the integrals involved in this Table, see:G(z) :-Table XXVII, p. 2877. F ( z ):-Table XXVIII, p. 2878. P(z):-J. Burgeea: Trans. Edin. Roy. SOC.,39, 257 (1899). S(z) and C(z):-Gilbert: MQm. cour. Acad. Bruxelles, 31 (1863); Ignatowsky: Ann. Physik (4)23, 894 (190j); Brit. Ass. Report, 1926, p. 273; Jahnke und Emde: "Funktionentafeln," p. 23, (1923). See also Table X X I I I and XXIV, pp. 2872 and 2873. F(a, v):TLegendre: "Exercicea de calcul int6gra1," t. 111, (1816); also Jahnke und Emde: loc. cit. p. 54. I For brevity in Nos. 2 and 3 of this Table, B is written for P: [odd] @cwhenf(u) = I Le. for (gz/8a&). G(4r2at/g2zz),and C for P: [oddl@cwhen f(u) = u, i.e. for the ex: presaion entered in the second line of this Table in the last column.

2816


TABLEXI1 (see p. 2818)

I($

No.

S:@c

I.

I

(XIIb,I)

2.

u

t.(XIIb,I)

3.

u2

t2.(XIIb,I)- 2t.(XIIb,2)

4.

eblu

e'' . (XIICI,I )

5.

b2u.e'"'

ebzt.{ b2t.(XIIa,I)- (XIIa,;)

6.

e-b2u

7.

b2u.e-b'u

- (XIIb,2)

e-'". { b2t.(XIIa,z) - (XIIa,8)

sin b2tfQ.(XIIa,3) - cos b2t

8. sin b2u+a

~

COS

)

(xIIu,~)

-

9.

+ 2(XIIb,3).

b2u+a

COS

b2t+a.(XIIa,3)

1

+ a.(XIIa,4)

+ sin b2t 4- a.(XIIa,4) 1

IO.

eKu.sinb2u+a

8'.(sin b2t+a.(XIIuJj )

- cos b2tfcu.(XIIa,6)

11.

eKU.cosb2u+a

eK1.{cos=.(XIIa,

+ sin b 2 t + ( ( X I I a J 6])

j)

TABLEXIIA The Sums of certain series which occur in S:@c and the P-approximations to their Sums Series = I.

No. F ( N )

2

-

I

N2a-b2

ZF(N).cos N g r 3.

2.

N

Finite expression for @(o) +ZF(2n).cos zngz and for Z F ( z n - I).COSzn- rgz

4.

P-approximation to expression of Col. 3

2n

-M.C1/S1

......

212-1

M.sl/cz

......

2%-

I

"(

- S3C3(C4C4+S4S4)+ 4 2 csS3(e,C,-s4S4)} I(C-4)

9,


2817

TABLE XIIa (Continued) I.

3.

2.

4

.....

In this table only, the letters have the following meanings:s = sin; c = cos; aZ = ( K 4 m )

+

= /

2

sinh; C = cosh; J f = a / q b d / a ; Q = a/4da(KL+b4) ; p = b?/2a; r = bgx/d/a; \E = W U + K

Writing f for s, S, c or C, the subscripts have the following significance:-

2818


TABLE XIIB

No. F ( N ) I

1

NZa

The Sums of certain series which occur in S : @c. Series = ZF(N).cos Xgx Z F ( 2 R ) . cos 2ngz Z F ( Z R - I ) . COS Z R - Igz 7i2 ngx 24a 4a 4a 8a 4a

-

- -ng5x5

480a3

+-.r440a3 96x6

TABLE XI11 The Original Integral (for use with approx. ui and zii). No.

f(u)

The Original Integral =

5.

Examples illustrating the use of Tables XII, XIIa and XIIb. These examples are necessary because of the compact form in which the information given in these Tables is presented; tables from which the results could be read directly, would take up too much space. Ezample (a):To find a finite expression for the Stationary Series arising from [even]*c when f(u) = ebZu. Look up f(u) = ebPuin Table XII; line 4 reads 5' = ebZi. ( X l Z a , I )


2819

which means S = ebzL.ZF( N).cos N g z with the value of F( N ) given in No. I of Table XIIa. Hence F ( . V ) = 1/(N2a b2) and therefore F(o) = Ijb2; so that S : [ e v a ] @ c= ebZt.Zcos Wgz/(N2a+bz), in which N is to be replaced by 212 since the series is of the [even]type. From Col. 3 of Table XIIa No. I it appears that, when N = 2n, Zcos N g z / (N2a+b2) = M.CI/S2-r/zb2; replacing M , C, and S I by the expressions for which they are abbreviations (given a t the foot of Table XIIa) the final result is

+

~ : [ e v e n ] @=c eb‘t.I;coszngz/(4n2a+b*)

(A)

Ezample (b): To find the P-approx. to the Stationary Series arising from [even]@c when f(u)= ebZu. Follow the procedure of Ex.(a) hut use Col. 4 of Table XIIa No. I instead of Col. 3. This gives

Ezample ( e ) : To find the Evanescent Series arising from [eoen]@c when f(u) = eb?u. The general rule is: In the expression for the Stationary Series (such as that given on the right of Eq. A , Ex. a) replace t by zero wherever i t occurs, and insert the factor e--.v20t immediately after the I; sign. I n the case chosen as example, this gives

V : [even]@c =

I ; ( e -4n*af.co~ zng2)/(4nza+b*)

Ezample ( d ) ] To find the finite expression for the Stationary Series arising from [odd]@c whenf(u) = I . Table XI1 line I reads S = (XIIb, I ) ; No. I of Table XIIb gives F ( N ) = I / A%; as the series is of the [odd] type, h ’ = zn-I and the last column of XIIb must be used. Therefore S = I ; c o s ~ . g x / ( ~ 2 = . ar2/8a-ragz/4a. ) Ezample ( e ) : To find expressions for S, V and P:S when the type of summation is neither [even] nor [odd] (see Table V, p. 2799). From the corresponding expressions for series of the [even] type those for series of the [all]type may be obtained by replacing a by a / 4 and gz by g x / 2 ;those for series of the [altl type by replacing a by a / 4 and gz by T - g z / 2 ; those for series of the [even-alt]type by replacing gz by r/z -gz. From the expressions for [odd]@cthose for [odd-altI@s may be obtained by replacing gx by r / 2 -gz. Thus (from Eq. B of Ex. a ) when f(u) = ebb

See. 8 :-Relations between Series‘ of the G E Group which differ only in the values of at and gx Representation of a series by a point:-Any series of the @ E group is fully defined by its symbol (if the form of f(u)and any constants contained in it be given) together with the numerical values of gz, a and t , or of gz, ~ ‘and 2 t. The value of gz and that of 42 may be given by the position of a point on a diagram such as Fig. 2 , and in what follows the series will be spoken of as being represented by the point; this implies however that the symbol, the form off(u) with its constants, and the value o f t (the same for every point on the diagram) are otherwise specified, for instance as a rubric to the figure. 1

Only series of the [even]and [odd] types are considered; see “Summary,” p. 2801.

W. LASH MILLER A S D A . R. GORDON

2820

N0.M TErms of V Iodd] [even] 2

08

6 6

8 8 16

16

32

32

0.2

9x FIG.2

-

In Fig. 2 , 0.227has been chosen as the ordinate of E; this gives “safe Boundaries” for [even or odd]+c and for [evenor oddlpc if quantities less than ~ o - ~ . Mmay t be neglected, and for [even or odd]@’c if quantities less than 8x 1o-~.M1may be neglected. The Scales of “Number of terms of V” are safe if quantities less than io-6.t may be neglected and if f(u)has one of the forms enumerated on page 2822.

Properties of the Diagram’ (a) If any line (such as OE, Ov, OF) be drawn upward and to the right from the point [.\/z=o, g z = o ] of Fig. 2 , the product2 &.(A:@c) will

have the same value for all series represented by points on that line. Approx. 2 8 2 6 , is based on this relation. (Proof:-This follows a t once from Eqs. 4 2 and 47, since g2x2/ahas the same value for all points on any line passing through the origin). ( b ) If any line (such as EB, vw, FC) be drawn upward and to the left from the point Q of Fig. 2 , the product &.(R,,:@c) will have the same value for all series represented by points on that line. (Proof:-This follom-s a t once from Eqs. 44 and 47, since ( x - g z ) 2 / a has the same value for all points on any line passing through [ d/at= 0,gz = x ] . ( c ) Similarly, &.(RIb:@c) will have the same value for all points on any line (such as BG) which passes through R ; &,(R2a:@c) on any line (such as GC) passing through S, etc. The positions of R and S are merely indicated in Fig. z ; their ordinates are d/at= 0 ; and their abscissas are - T and ZT respectively. ( d ) If the line from 0 and the line from Q intersect on the ordinate passing through P, as a t the points E, v and F of Fig. 2 , thevalue of ( A: @c) charac-

ii, p.

+

.\/a.

can be used as it stands for series of the * E Group (see p. 28 j6) as well as for series oy‘khe @E Group. If used for W E (see p. 2834), 1 0 - 6 . M t must bereplaced b y 8 X r 0 - ~ . M t (see p. 2835) but IO-O.CC~remains unaltered. * T h e quantities &.(A:@c), &.(RIo: @c) etc., are those which arise in evaluating Solutions 3 and 4 of Table I.

Fi

2

S U M E R I C A L EVALCATION O F INFINITE SERIES

282 I

teristic of the first line will be identical with the value of &.(R,,: @c)characteristic of the second. (Proof:-This follows a t once from Eqs. 42, 44 and 47, since when g x = x / z , ,4= R J . Appror.iii, p. 2 8 2 7 , is based on this relation. Similarly, if a line through Q cuts a line through R on the ordinate gz=o, the value of y‘a.(R,,:@c) characteristic of the first line will be the same as the value of &.(R,b:@c) characteristic of the second (see Eqs. 44, 45 and 4 j ) ; and so on. Thus 4;. A on OE = R,, on EB = 4; R. r hon BG = &R,, on GC, etc. Boundaries I t is now convenient to discuss the properties of the set of lines OE, EB, BG . . (Fig. 2 ) which result when t’he ordinate of E is so chosen that the absolute value of R,, for the series represented by that point is just negligible in view of the accuracy desired. In the first place, it is obvious that R,, must be negligible a t all points on the line EB; since as has been shewn above, d/a.R,, is constant for all points on that line and 4;increases from E to B. In the second place, if at be not too large,’ and if f(u)be one of those which retain their signs (which without loss of generality may be assumed to be positive) while u increases from o to t , then R,, will decrease if a be decreased while t and gz remain una1tered;so thatif R,,is negligible along the line EB, it must be negligible a t all points below it. Hence for functions of this kind, the line EB may be spoken of as the “Boundary* for R,,.” Similarly, BG is the Boundary for R r b ,GC for R,,, etc.; but as OE runs downward (Le. to smaller values of di)from E, this line does not constitute a Boundary for A :@c. For all practical purposes, moreover, whenever the line E B is the Boundary for R,,, it may also be taken as the Boundary for Rem:@c;for the sum (R,,+RIb) a t B is just equal to R,, a t E, while a t every point on EB, R,,, Rzb,etc., vanish in comparison with Ria. From this it follows that appror.i, which assumes that P:@cis practically equal to @c, may be employed only when the point representing the series lies on or below EB. R7hen the ordinate of E is fixed, the Boundaries can be drawn with ease;3 this ordinate gives the value of d/at for which R,,:@c is just negligible when gr = x / z . Table XIV gives the ordinates of E for [even or o d d ] @c when f(u) = I and quantities ranging in magnitude from ~o-’O.t to 10-~.t are the largest considered negligible.? The only form of f(u) other than ~ ( z L = ) I for

.\/a.

For a(RIa : [even or odd] .Pc) I J a to he positive when f(u)is, positive, it is sufficient (see Eqs. 44 and 47) that (a-gz)2> z(at-au), i.e. at < a2/8. Similarly, 8(RIb)/aa n.ill he positive if (T gz)2 > z(at-au), i.e. at < x 2 / z . I n other words, the qualification in the text is equivalent to “if the ordinate of E is less than i.e. is less than 1 . 1 I . ” 2 I t would he an ezact Boundary for .\/;.Ria, which is the quantity that arises in evsluating Solutions 3 and 4 of Table I, if the ordinate of E were chosen so that 6. R I a~t gz = was just negligible. The ezact Boundary for RIa itself coincides with E B a t E and else. this where lies slightly above it; thus, strictly speaking, EB is a safe Boundary for R I ~but adjective is kept for Boundaries that are safer still. 3 From their definitions, the ordinate of B is twice that of E, the ordinate of G is three times that of E, etc; so that OE and BG are parallel, and so are EB, GC, etc. [even or o d d ] * c ) / t = ( 6 / 1 6 a t ) . G ( 1 6 a t ) ;see 4When f(u)= I . and gz = a / z , ) the values of at given in Table XXVII Table XXVI No. 2. The values of Zog ( R I B / t for were computed and plotted against &; whence the ordinates of E given in Table XIV were found by interpolation.

+

m,


2822

which the integrals needed in the construction of such a table have been tabulated is f(u)= u p ;but in the paragraph on “Safe Boundaries” it is shewn how the ordinates of Table XIV may be made generally useful.

TABLE XIV Ordinates of E for [even or odd]Gc when f(u)=

RJt

10-l~

d/at

0.179

I.

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0.185

0.194

0.208

0.227

0.252

0.288

0.342

Fig. 2 is constructed on the assumption that when f(u) = I , the quantity 10-0 !is just negligible; if ten-thousand-fold greater accuracy had been required, the point E would have been lowered only 20%; if however quantities as great as IO-3.t can be neglected, the point may be raised 507~.

“Safe” Boundaries:-By Rule I, p. 2840, lR,,:+cl < M . ( R I a : G c ( ~ )where ] Ai’ is the maximum absolute value attained byf(u) in the interval o < u 10-12, and all the other terms of the Remainder are smaller still. Hence P:[odd]*c- Rem:[odd]@c=0.00076 0 1573, accurate to the tenth decimal. T o attain this accuracy by the use of approz. u, the Stationary Series and eight terms of the Evanescent Series must be computed and added.

Approx. ii, iii and iu. When the point p representing the series to be summed lies below BE of Fig. 2 , Rem:@c may be neglected; so that [euen]@c may be replaced by A : [ e c e n ] O c - frLy(u).du, and [oddlac may be replaced by A:[odd]iPc,see Eqs. 41 and 46. In approx. i it is assumed that the value of A : @ c for the series represented by the point p can be found from tables; in approx. ii, izi That is, whenever the integrals have been tabulated. The ordinate of E in the figure a s printed is fixed by the assumption that xo-B.Mt is the largest quantity that may be neglected. other assumptions lead to other ordinates for E , see Table XIV. If the point representin'g the series to be summed lies on or below BE, the error of approz. i cannot exceed the quantity assumed negligible in fixing the ordinate of E.

2826


and i v it is shewn how its value may be computed from the sums of certain “auxiliary series,” such as those represented by the points s, v and w respectively in Fig. 2 . All these approximations are based on the relations established in ( a ) and ( d ) of p. 2820, viz.1

[&.A:@c], = [ ~ L A : @ c=] .[d/a.A:@c], = [&.R,,:@c],, since p s and v lie on a straight line passing through the origin, and w lies on Qv. An expression is given for the error2 of each approximation, in this case the algebraic quantity that must be added to the value given by the approximation in order to secure the true value of the series. For lack of tables it is in general impracticable to compute the numerical values of these expressions, but approximations to their absolute values are given in terms of the quantity assumed negligible in fixing the ordinate of E; as shewn in the Examples, closer estimates may be arrived a t by means of Rule I, p. 2840. I n the case of approx. iii and iv, the constants of the auxiliary series depend only on those of the given series; and if the upper limit found for the error is too high, these approximations must be replaced by others. In the case of approx. 2 2 , the constants of the auxiliary series depend also on the position of E, and although a slight lowering of E may have but little effect on the value of h (see Eqs. 49 and so), Table XIV shews that it will greatly decrease the absolute value of the other factor of the error, viz. [Rem:@c],. A p p r o x . ii:-At the point s where any line Op from the origin cuts BE, Rem:@cmay be neglected (see Boundaries, p. 2 8 2 1 ) ; so that a t this point @c-+P:@c.Hence by means of ( a ) , p. 2820, and Eqs. 4 1 and 46,

[even]

+

The quantity h, viz. the ratio [ & I 8 / [ - \ / & ,is given by h = 2 Ea/(a.\/Z ~ E g x in ) which E is the ordinate3 of the point E in Fig. 2 , and .\/at and gz are to be given the values they have in the series represented by the point p. To make (49b) and ( s o ) exact, the quantity h.[Rem:@c],- [Rem:@c], must be subtracted from the right of (49b) and added to the right of ( s o ) . I n estimating the error introduced by neglecting this correction, [Rem:@c], may bereplaced by [RI,,:@c]~, and (unless p is very close t o BE) [Rem:@c], may be neglected; that is, the error of approx. ii is about h times the quantity assumed to be negligible in fixing the ordinate of E. In this notation the values of 4i and gz are indicated by the subscript, all other constants are the same in the four series. * If p lies near the ori in, [Rem:*cls will as usual be negligible; but h will be large and so may be the errors of a f three approximations. For such cases however the percentage error in [A:*cIp will be the same as the percentage error in [ A : 9 c l aor ”, or in [RI,: *C]W as the case may be. In Fig. 2 , as printed, E = 0.227;but any other value might be chosen, see Table XIV.


2827

The relations of Eqs. 49 and 50 hold for any point below s on Op, i.e. for values of h smaller than those given above; but the number of terms of V needed in evaluating the auxiliary series is least at s. Approx. iii:-At the point v where Op cuts PH, A:@c = RI,:@c,see ( d ) p. 2 8 2 0 ; and if v lies below GI RI,:@c+ Rem:@c,see Boundaries, p. 2821. If the series be of the [even]type, Rem:@c= @c - A:@c i x ' j ( u ) . d u ,see

+

-

Eq. 41; hence if p lies below OG, [A:@c],+[@c A:@c],+ 3Joff(u).duI i.e.

2[A:@c],+ [@c],

+ ; l ' f ( u ) . d u ; and by ( a ) of p.

[even]

2820

4

where h = n/2gx in which gx is to be given the value it has in the series represented bv the point p. To make ( S I ) exact, [Rem:@c], - (h/z).[Rem:@c R,,:@c],, or in prac- (h/a).[R,,:@c],, must be added to the right. The absolute tice [RIa:@cIp value of this correction cannot reach ( I h / 3 ) times the quantity assumed to be negligible in fixing the ordinate of E. Similarly,

-

+

+ To make exact, [Rem:[odd]@c], + (h/2).[Rem:[even]@c- Ria: [eeen]@c], or in practice [RI,:[odd]@c], + (h/2).[RIb:[even]@c], must be subtracted from the right. The absolute value of this correction cannot reach + h / 3 ) times the quantity assumed to be negligible in fixing the ordi[ [ o d d ] ~ c ] , - +(h/n.[[ecenl@c], (h/?).J>(u).du 0

(52)

(j2)

(I

nate of E. Approx. io:-At the point w where the line drawn from Q through v cuts OC, [&R,,:@c], = [&.A:@c],, by (d) p. 2 8 2 0 ; and [R,:@c],= z[R,,:@c], since w is on the ordinate where gx = 0,see Eqs. 45 and 47. If w lies below C, R,:@c-* Rem:@c = i (@c - P:@c)all for the point w; the minus sign is to be taken when the series is of the [odd] type, see Eq. 46. Therefore if p lies below OF, [even]

[odd1

- P:@c], [@c],+ (h/a).[P:@c - @Clw [@c],+

(h/2).[@C

$J>(u).du

(53)

0

(54)

where h = n/gx in which gs is to be given the value it has in the series represented by the point p. To make ( 5 3 ) and (54) exact, [Rem:@c], =F (h/2).[Rem:@c - R,:@c],,or in practice [RIn:@c], - (h/2).[R2:@c],, must be added to the right of ( 5 3 ) and subtracted from the right of (54). I n the worst case the absolute value h / 4 ) times the quantity assumed to be negliof this error cannot reach ( I gible in fixing the ordinate of E. If p lies below OE, w will lie below B, [@c - P:@c],will be negligible, and Eqs. 53 and 54 become

+

[even]

(55)

W. L A S H MILLER A N D A. R. G O R D O N

2828

[+cl,--t

[odd1

0

(56)

+

The correction to be added to the right of ( 5 5 ) is practically [RIo:+cIp (h/z).[R,:k],; the correction to be added t o the right of ( 5 6 ) is practically - [Rla:@cIp (h/2).[R1 :+c],; in neither case can its absolute value reach (I h/z) times the quantity assumed to be negligible in fixing the ordinate of E.

+

+

Approz. ii. Ezample 4:-To evaluate [odd]Ocwhenf(u) = sinb-, t=o.oj,a=o.o6~j,gz=n/160, b 2 =jon, n = n / 6 . The given series is represented on Fig. 2 by the point [&=O.Oj6, gz = a/160], which lies well below BE. The line drawn from the origin through this point cuts B E a t \& '0.43, so that a safe value' for h is 0.43/O.Oj6 = 7.6, Le. if h be given the value 7.6, the quantity neglected in (50) will be less than Io-6.hAVft=1 o ~ x 7 . 6 X 1 . o X 0.05~ 3 . X8 10-7. The actual computation was made with h = 8, i.e. the constants for the auxiliary series were t=o.oj, a = 4 , g z = n / z o , b Z = j o r , a = r / 6 ; Rule I shews that the quaqtity neglected, viz. 8 times the Remsinder of the auxiliary series, cannot exceed Setting h = 8 in (jo) gives 0.1 12463 as the sum of the given series; its true value, 6 X IO-'. obtained by evaluating the expression for S given in Table XI11 No. 8 and computing and adding 33 terms of V , is 0.1 12463. In evaluating the right hand member of (jo), Oc was replaced by S - V , see Eq. 16; four terms of the auxiliary Evanescent Series were needed instead of 33 of the Original Form. A further considerable saving of labour might be effected by computing the Papproximation to the auxiliary Stationary Series instead of the auxiliary Stationary Series itself; this substitution' is advantageous because of the relatively simple form of P : S ; it is allowable in the present case because with the given values of the constants a, gz and b2 and the value taken for h, the sinh y and cosh y of Sec. Iob p. 2837 are each equal to, eY/2 within two parts per million. With this substitution the result is 0.1 12463.

Ezampk 5 : T o evaluate [odd]Oc when f(u)=eKU.sZnb K a , t = o . r , a=o.o9, gx=o R = O . Obz=o.Iz, ~, a - n / 4 . This series is represented by the point [v6t=O.O95, gz=o] on Fig. 2, which is well below BE. Hence if h =0.4jj.i0.09j=4.8, the quantity neglected ~ . value ~XIO chosen - ~ ; for the in (jo) will be less than ~ o - ~ X ~ . ~ X ( M = ~ . ~ ~ ) X O . I =the computation was h = j , so that the constants of the auxiliary series have the values: t =o.I, a = z . z s , gz=o, K = o . o j , b2=0.12, n = n / 4 , and by Rule I the error (viz. 5 times the value Eq. jO gives j(0.3702 776 of the Remainder of the auxiliary series) cannot exceed I .o X -0.2366 696) = 0.668040 as the sum of the series; to obtain this result it was necessary to evaluate the expression for S given in Table XI1 No. I O using the constants of the auxiliary series, and to compute and add 4 terms of the auxiliary Evanescent. The true value of the series, obtained by computing S and 1 7 terms of I-, is 0.668041. I n this example, with the numerical values assigned to the constants, the substitution of P : S for S in evaluating the auxiliary series is inadmissihle, because of the errors introduced by replacing the hyperbolic sines and cosines by single exponentials (see Ex. 4). A p p r o x . iii. E z a m p l e 6 : T o evaluate [oddlac whenf(u)=e-b?U,t=o.o18, a = 1 / 9 , g x = r / 3 0 , b2=z.o. The point p representing this series on Fig. z lies within the triangle 0 JH well below BE. I n Eq. 5 2 , h = n/zgz = I j ; so that the constants of the auxiliary series are: t =o.or8, a = 2 j , g z = n / z , b2=z.o; as p is well below BE, R,,:[odd]*c is completely negligible, and (from Rule I) 15 times R1t:[euen]9c for the auxiliary series cannot exceed 6x10-8, which therefore constitutes an upper limit to the error of the approximation. I n evaluating the right

' See the expression for h given after Eq. jo. If this substitution be made, the quantity to be subtracted from the right of (49) and added to the right of (so) will be h. [ R a : @ c- Rem:S],; so that in certain cases, the approximation may act,ually be improved by the substitution.

SCMERICAL EVALUATION OF ISFINITE SERIES

2829

of ( 5 2 ) , +c was replaced by S - l - = -0.0081205+0.0016848= -0.0064357, see Table XI1 S o . 6; lvhence ~ . ~ ( - 0 . 0 0 6 ~ ~ j ~ $ 0 .=o.o18032 0 0 ~ 8 j ~for ~ ~the sun] of the series; t \ w terms of the auxiliary Evanescent were required. This series \vas also summed by the use of n p p r o z . ii: using h =- 7 . 5 which is very close to the best value; five terms of this auxiliary Evanescent were required; \).hen the Original Form { a p p r o x . c ) \v:i~used, 34 terms of the Evanescent were needed; in h t h rases the i-esult was the same as that obfained by a p p r u z . iiz. A p p r o x . ic. Ezample 7 : T o evaluate [oild]*c whenftu) t=o.oz. a - I , gx= series is represented on Fig. z by the point [ c , % = o , I ~g, I = r , 51, which 1 iyell below BE. In Eq. j 4 , h = j;so that t!ie ranstants of the auxili:lry ser a = z j , g r = o , b ? = ? ; by Rule I, the error (viz. 2 . j times X : : @ c for the ausilinry~is less than g x 1 0 - l ~ . Employing the expression for P : [ o d d ] + cgiven in Tnble S I S o . 6 and those for S and 1-given in Table XI1 S o . 6, Eq. 54 gives 2.j:o.oz.+.+oXqz-o.ojo8oo4z -0.02642072) = 0.0000718 as the sum of the series. Three terms of the auxiliary Evanescetit vere needed. The use of npprox. ii would require the coniputvtion of a Stationary Series and seven terms of an Evanescent to attain an acrurcrcy of ? X 10-6. The com;'iitntion was also carried out hy means of the Original Form iapproz. ti;, the Stationary and tn-elve ternis of the Evanescent being required to ensure accuracy in the seventh decimal place; the result was . the same as that obtained by o p p ~ o z iv.

[Ezample R:-To evaluate [even]%c whenfiu) =~/fl+ lies below OFof Fig. z and a little above BE. In Eq. 53,h = z . j ; so that the constants o!' the auxiliary series are: t =o.36, a = I . j625, gz =o, b? = 1.44. As (by Rule I) 1.2 jiR2:Gc) for the :tiixiliary series is less than IO-", the only quantity that need he considered in estimating the error is R,,:@c for the given series; by Rule I this cannot tweed z ;A: IO-^; it is positive and m u s t he added to the approximate value. I t is not possible to evaluate %c for the auxiliary i~vhichappears on the right of Eq. 53) by the use of the Original Form ( a p p r o z . C J , since the series belongs to Class I1 (see Table S; and there is no finite expression*for S , but in Ex. I i the sum of this s he 0.187839 by the use of uppros. tli; the value of P : @ cfor the au:iili i1:1iso.rt;p76;sothat thesumsought is 1.25~o.1SjY39-o.ib;o;6) -0.141641 = -0.140687, to which something less than + z X ~ o -must ~ be :idJPd. The series of this example n-as also evaluated by means of appror. ir, whirh gave -0.1.+0686 exact to the last place; this method of evaluation is preferable to that described above, as it aioids the complicated computations of Es. 11.1

A p p r o x . i', c i z . the Originni Forvc. The relation is @c = S - T7, see Eq. 16, p. 2803. Es. (1 shews how convenient this approximation may be with favourable valucs of the constants, Ex. I - j how inconvenient with others. When f(u) = e'"', si): or COS b2zi, or eKu.sin OT COS b'u, P : S may sometimes be substituted for S {sre Sec. IO^, P. 2 8 3 7 ) . If j ( u ) have one of the forms listed under D or E in Table S,approx. u may be used in spite of the fact that no finite expression is known for S , because in these cases an approximate value for S may be obtained by the method of See. IOC, p. 2837 (see Ex. I O ) ; but this procedure usually involves more labour than does the evaluation of @c by approx. ci or vii.

S might be evaluated by the method of Sec. ioc, illustrated in Ex. 1 7 , but the computation is extremely laborious.

2830

W. LASH MILLER AND A. R . GORDON

Ezampk 9:-To evaluate [euen]@c when f(u) = &*IL, t = I , a = I , gz = r / 8 , P =&. From Table XI1 No. 4, [e!,en]@c= S - V = 0.620782 - 0.002159 = 0.618623. Only one term of the Evanescent Series is required to ensure accuracy in the sixth decimal place. [Ezample IO-To evaluate [even]Qcwhenf(u) = I / & , t=r.44, a=o.25, g z = z r / j , b2 =0.36. The expression for V may be found from the Original Integral entered in Table XI11 No. 2 by setting N = z n and u = o in the latter, multiplying the resulting expression bycoszngz,andprefixingthe Z sign. It is V = ( r / d / e ) . z ~ o s z n g z . e - 4 ~ ' ~ ~ . F ( ~ q n Z a=b ' ) / n -0.181 j r o (three terms needed); evaluation of S by the method of Sec. roc gives S = -0.695060, see Ex. 1 7 ; hence Q c = -0.513550, exact to the last place. The method of approz. vii (seeEx. 12) is much to he preferred, because of the labour involved in evaluating S by the method of Sec. roc.]

A p p r o x . vi, vii and viii. From the definition of the +-Family given on p. 2795, and those of P : 8 c and Rem:& given in Eq. 2 7 - 3 1 , there follows

A p p r o x . vi, vii and viii are based on Eq. 57b in which r is to be given the smallest value that will permit the third term on the right to be neglected. Third term of Ep. 57b:-An upper limit to the absolute value of this term

it

is obviously given by M * .

(Rem:Bc).duwhere M* is the maximum absolute

value attained by f(u)in the interval r < u < t . In this the integrand is of the form F(t-u).du, see Table VIII; so that by changing the variable from u to T + U , the expression for the upper limit can be put in the form

M * . [&-r-u).du

= M*.(Rern:+*c(I)] where +*c(I) isaseriesidenticalin Jo all respects with +e except that in it the t and f(u)of +e are replaced by t - r and unity respectively. Thus if r be so chosen that .\/a(t-lies .) on BE or below it, the error introduced by neglecting the third term of (57b) cannot exceed ~ o - ~ . M * . ( t - r )assuming , that the ordinate of E is that of Fig. 2 as printed; and therefore must be less than I O - ~ or Xin~general , than the quantity assumed negligible in fixing the ordinate of E. The best value of r is found by equating .\//a0 to the ordinate where BE cuts the perpendicular drawn downward from the point (such as p' of Fig. 2 ) which gives d/at and gx for the series to be summed; because lower ordinates give larger values of r , and the labour of evaluating the first term of ( 5 7 b ) increases with increase in r (see below). Second term of Eq. 57b:-Once r has been chosen, the value of the second term of (57b) can be found a t once if tables of the Approximation Integral A :9 c have been constructed (approx. vi, uiiz); if they have not, computation by tabular integration (approz. vii) is best, for although evaluation by means of approx. iv is sometimes possible (see Ex. 13) it involves more labour. First term of Eq. 57b:-If the Original Integral has not been tabulated, the first term of ( s i b ) must be evaluated by tabular integration (appror.

XUMERICAL EVALUATION OF INFINITE SERIES

2831

viiij. If the Original Integral has been tabulated, the term in question may be replaced by the difference between V :@c and a new series which also contains a converging exponential factor (approx. vi and vii), thus :-

which follows a t one from the expression given a t the head of Table XI11 for the indefinite integral. Expressions for cp( ~ V , Tmay ) be obtained from Table XI11 by writing 7 instead of u ; this table deals only with series of Classes D and E, but these are the only Classes for which approx. vi and vii are recommended. For use in See. IOC,it may be noted that the first series on the right of (58) becomes S:@cif T be replaced by t ; the greater f - - 7 , the more rapid the convergence. A pprox. vi. This approximation assumes that the first term of ( 5 7 b ) can be evaluated by (58); also that A:@c has been tabulated, which in practice means that gx = 0 . Example ff:-To evaluate [men]@cwhenf(u)=I/.\/b2+U, t=o.36, a = 1 . j 6 2 5 , g z = o , b* = 1.44. From Eqs. j 7 b and 58, by means of Table XI11 No. z and Table XI No. I I , [ a r a ! @ c-+ ~ { e - 4 ~ * ~ .(F~ (- ~z )n d G ) F(zn-) I/nfi ( d ; / ~ . \ s/i& n - '.d ( t - r ) / ( b ' + t ) .\/6 The best value for T (see p. 2830) is given by 0.455 = s r , whence T =0.227; the larger value r =0.3024 was chosen (although this necessitates the computation of two more terms of the first series on the right of Eq. 58) in order that damight be a bo-figure number and interpolation in Table XXVIII might be avoided. The value i=0.227 corresponds to a possible error of ~o-'j..M(t-~)=IO-'; with ~ = 0 . 3 0 2 4 ,Rule I fixes the error as leas than

+

+

10-12.

Setting

=0.3024, the preceding equation becomes [evenl*c -+ o . o ~ ~ o o ~ - 0 j036+ .01 Five terms of the first F series and two of the second had to be computed and added. T

0.127513-1.341641+1.32oooo=0.187839.

Approx. vii. This approximation is identical with approx. vi except that tabular integration (approx. is) is employed in evaluating the second term of (57b). The method illustrated in Ex. 13 is, strictly speaking, an independent approximation; in it the first term of ( j 7 b ) is evaluated by (58) and the second by the method of appror. iv. As this procedure involves more labour than does approx. vii for the only Class of series with which it might be used, no number has been assigned to it. Example 18:-To evaluate [menl*c when f(u)= I/-, t = 1.4, a=o.25, g x = 2 = / 5 , h2=o.36. From 0.27=-, r=1.148; with this value leas thano.gX~o-eisneglected (see p. 2822). The convenient value ~ = 1 . 0 8Was chosen, which ma%= -=0.6. With this value the quantity neglected in (j7b) is leas than z X 10-6 (see Rule I); as f ( u ) and Rem: t)c are positive, and the series is of the [men] type, the quantity neglected is positive. Eqs. 57b and 58, and Table XI11 No. 2 give

K. LASH MILLER ASD A . R. GORDOS

2832

Five terms of the first F series and three of the second were needed, their algebraic sum is -0.372865; the value of the last expression on the right of ( j g ) is -0.141641. The integrands of the remaining: expression on the right of (jg), viz. (\/;'z) , e - o . , m ~ , 1 4 ; - , < ! / v i i . + - i ~ ( o . 3 6 J - u 1 I

xere computed1 for u = I .OS, u = 1.10, . . 71. = 1.44 and differenced; taking account of third difterences the formula of Ex. 15 p . 2833 gives o.ooo9jj as the value of the integral. Hence [eceii]@c + -o.37286j o.ooo95j - 0.141641 = -0.513jj1, to which something less than zX1o-6 must be added. Evaluation of the Original Form (Ex. I O ) gives - o , j x ~ . ; j o exact to the last place, but is much more laborious.

+

jE:samplc 13:-To evaluate [ecen]@cwhen jmuj and the constants are those of Ex. 12. Setting T = 1.08 as before, the values of the first and last expressions on the right of ( 5 9 ) are found as in Ex. 1 2 ; the remaining expression is identical x i t h the .i:[everL:@c of E x . 8 and may be evaluated by the method of nyprox. it, ns in that example. The result is the same as that obtained in Ex. 12, but the method is not to be recommended; for although in Ex. 1 2 fourteen values of the integrand had to be computed, the operations involved are far simpler than those of Ex. 8 and accidental errors are immediately apparent; the method of Ex. 12 is far preferable to those ot Ex. I O and Ex. 1.3.)

dpprox. L This approximation is identical with approz. ti except that the first term of ( j i b ) is to be evaluated by tabular integration (npproz. iz) instead of by the method of Eq. j 8 . The assumption that *4:+e is tabulat'ed restricts the use of this approximation in practice to series in which gx = 0. Example 14:-To evalu [odd]@c \vhen ~ 7 u=) I Id( V + u , :c2 f u I , t = 0.7, a = I , gx =o, 6*=1.;, c2=o.j, as accurat as four-place tables of the First Elliptic Integral permit. The error cannot reach IO-* t - T ) if 0 . 2 8 8 he taken as the ordinate of__ E (see Table XIV); assuming this ordinate, the best value of T is given by z E =o.576 = t'o.; -7, ;.e. T =o.368, the error corresponding t o which is less thrin o.4X 1 0 - 4 . For convenience in rhe use of tables, the (safer) round number T = o . ~was chosen; with this value the error is less than 1 0 - 6 (see Rule I). With T = o . ~ ,the second term of ( j i b ) becomes I X ' T 9.6i.F(45', 30") = 0.3064, see Table XI S o . 14. To evdu:ite the first term of i . i i h ~ ,a tahle of the integrand 7 -U\, 1 / . , / ~ , > ~ ~ ~ ~ , ~ j [otjd]oc,j(u) = e - .ti 7 - t i ' l.c -!) :O i -d + - 2 5 I[) ,

\vas computed and differenced ior the nine vilues u = 0.00, O . ? j , 0.10 . . . 0.40. Taking account of third differences, the formula of Ex. 15, p. 2833, qives 0 . 2 1 jS as I.>< value of the integral; hence [ o d d j Q c i0 . ~ 1 7 8 0.3064 = u.5242 exact within one unit in the fourth decimal place.

+

dppi.0". is.

Khether or no any of the other approximations are available, any series of the @ E Group can be evaluated by tabular integration; in Ex. 15, the form of j ( u ) is such that none of the other approximations can be used. T o carry out such an integration, a table of the values of the integrand Bc.j(u) must be computed and differenced (the values of Bc being found as shewn on p. 2 8 0 9 ) ; since Bc changes w r y rapidly in value with change in u when u approaches t , it will usually be advantageous to divide this table into two, in the first of which (values of u near zero) the interval between successive values of u is greater than it is in the second (values of u near t ) . TVhen the For u 2

I .36

their values xere negligible.


2833

table or tables have been set up and differenced, the integration is effected by means of the formula’ given in Ex.I 5 . Ezampk f6:-To

evaluate [odd]& when f(u) = 1/(b*+t2), t = 0 . 2 4 , U = I , 92=a/4, With these values of the constants, [oddlec may hereplaced by A : [ o d d ] 8 e - R I , : [oddlecwithout introducing an error greater than 10-7. The following table gives the values . O . I Z and their differences; the factor &/4 is taken of the integrands for u =o.oo, 0.02 out to Rave labour. bZ=o.j.

..

e-a2/64(0.24--u)

( 4 1 4 ) . [oddIec.f(u)

.06

F(u) 2.1346 2.1059 2.0575 1.9849

.08

1.8822

.IO

1.7415 1.5523

U 0.00

.02

.04

0.12

A’!

A’

-0.0287 ,0484 ,0726 ,1027 .I407 -0.1892

-,-9n2/64(O.24-u)

= F(u)

( d F .iO.j+U*)

-+

-0.0197 .0242

,0301 .0380 -0.0485

A‘‘‘

Aiv

-0.0045 ,0059 ,0079

-0.0014 .0020

A” -0.0006

-0.0006

-0.0026

-0.0105

13.4589 The value of the integral is computed from the 8um of the entries in the second column and from the first and last entries of the last six columns of this table, as shewn in the tollowing equation: F ( u ) . d u = 0.02{ 13,4589 - ‘(I.j523+2.1346)

- 1(-0.1892+0.0Z87) I2

- L ( - 0 . 0 4 8 5 - ~ . ~ ~ 9 7 ) - -I9 ( - o . o I o ~ + o . o o ~ ~ ) - -(3 -0.0026-0.0014) 24 720 I 60

___ 863 ~-0.0006+O.OOO6) 60480 Here 0.02 is the common difference of the numbers in the first column, while the coefficients I / P , 1 / 1 2 , 1/24 etc. are deduced from the theory of interpolation, and are the same for all problems of tabular integration, without regard to the tabular interval or to the form of F(u). Asimilar table of the integrands for u = O . I Z , 0.13,. , 0 2 4 , g i ~ e s / ~ . ~ ~ F ( u )=O.O76j49. .du 0.12

+

Hence [oddlac + (d;/4) . (0.232638 0.076549) = 0.13701, exact to the fifth decimal place. A more exact computation in which F ( u ) was computed to the i f t h decimal place for u=o.oo, 0.01, . . . 0.19 and for u=0.190, 0.195 . . 0.240, and inwhichfifthdifferences were taken into account, gave [odd]& --t (&/4) (0.3048098 0.0043806) = 0.137007, exact to the sixth decimal place. .4 rougher computation in which F ( u ) was computed to the third decimal place for u =o.oo, 0.03, . . . 0.24, taking account only of first differences, gave [odd]*c = o.13;1, a result which would be accurate enough for many lahoratory computations.

. .

+

Procedure to be followed when f(u) is discontinuous Discontinuous forms of f(u)occur in computations of the concentrationchanges due to electrolysis with commutated or otherwise abruptly varied current. Let f(u)have the successive values2:fi(u) from u = o to u = tl, See Boole: “Calculus of Finite Differences,” 3rd Ed., p. 55 Eq. 28, (1880); also Rice: “Theory and Practice of Interpolation,” p. 136 Eq. 235, (1899). * The functionsfi(u),f*(u)are by definition continuous within the ranges specified.

2834

W. LASH MILLER AND A . R . GORDON

JP(u)from u = tl to u = t z ,

. .

. f J u ) from u

=

tT-I to u

= t, = t .

Then

When the integrations have been performed, Eq. 60a gives @c as the sum of a number of Evanescent Series plus one Stationary Series (arising from the last integration); this is the most convenient form when the intervals t - t ~ , t-t2, etc. are large. Eq. 6ob expresses @c as the sum of a number of series of the @ E Group; this is the most convenient form when the intervals are small, because it enables approx. L-iu to be employed. A number of special cases are dealt with by Rosebrugh and Miller: loc. a t . , p. 834-846 and 868-875.

Approximations to the Derivaties of

with respect to gx

Such series occur in Solutions Nos. I and 2 of Table I ; the methods developed for evaluating are available with certain modifications for these series as well. Symbols:-For brevity, @' will be written instead of d @ / d ( g x ) . I t can easily be verified that d ( P : @ c ) / d ( g x ) = d ( S : @ c ) / d ( g x ) , and this will be written A : @ ' c ; from Eqs. 42 and 46, it follows that

Similarly, R r a@'c : will be written for d(R,,: @ c ) / d ( g x ) ,etc. W h e n gx=o:-By differentiating the expressions for R,,,: @c and Rnb:@c given in Eqs. 44, 45 and 47 and then setting gx = 0 , it can be shewn that when g x = 0 , Rna:@'c = - Rnb:@ I C i.e. R,,:@'c = 0 , whence Rem:@'c = o and therefore @'c = A : @ ' c ; this eliminates upprox. i t a , see p. 2836. As shewn in the foot-note to Table I, when gx = 0 , [euen]@'c = (odd]@'c = a f ( t ) / 4 a ; the same results can be obtained by replacing the variable u by y = g z / z d a t - u u in the expression for A : @ ' c given in Eq. 61,and then setting gz=o. W h e n gx = */&:--It is apparent from the Original Form that when g r = a/zJ [even]@'c= 0 . By differentiation of the Poisson Form, it can be shewn that when g x = * / 2 , A : [even or odd]@'c = - R r a [euen : or odd]@'c, a relation which is used in A p p r o x . iiia, see p. 283 j.


2835

Boundaries:-From the expressions for A :W c given under “Symbols” above, it follows that’ a.(A:@ I C ) has the same value for all series represented by points on any line passing through the Origin of Fig. 2 ; similarly’ a.(RIa:@’c) has a constant value along any line through Q; etc. From this and the special relations when gx = o or ~ / 2 ,it follows that a.(A:@’c)on OE = - a.(RIa@’c) on EB = U.(R,bl@’C)on BG = -a.(R,,:@.’c) on GC etc. Ordinate of E:-The values of -\/at for selected values of R r a :[evenor odd] 2 given in Table XVI. These are the @IC when f(u)= I and qz = ~ / are ordinates of E for W c when f(u)= I and the quantities ~o-’O.t, . . . 1 0 - ~ . t are regarded as negligible. I n Fig. 2 as printed, the ordinate of E is 0 . 2 2 7 ; this corresponds to neglecting ~ o - ~ . Mwhen t the Boundaries are those for [even or odd]@c, and to neglecting 8x1o-~.Mtwhen the Boundaries are those for [evenor odd]Wc. TABLE XVII Ordinates of E for [evenor odd]Wc when f(u)= I .

R,,:Wc/t

dat

IO-~Q

10-9

IO-*

10-7

0.167

0.173

0.181

0.193

10-6

10-~

IO-^

0.229

0.257

0.300

10-6

0.209

,Vumber of terms of V:-As the differentiation introduces an hi into the numerator of each term of the Evanescent Series, V:Wc will be less rapidly convergent than V:@c;but it is nevertheless more rapidly convergent than the series on which the Scales of “Number of terms of V” of Table XV are based; these therefore remain unaltered. Errors of the approximations:-The equations given below are exact, the quantities neglected in the approximations are printed within { ] and will be referred to as the “error”; upper limits to their values in any given case may be found by Rule I, in which @c is to be replaced by Wc. Approx. ia:-

0, Rem: P c [@’c], = h2.[@’c],i (h2.[Rern:3’c], - [Rem:@’c],)

W c = A : W c i (Rem:@’c].When qx =

Approx. iia:-

=

0.

the plus sign to be taken when the series is [odd]. Here as in Eqs. 49 and 50 h2 is the quotient of the value of a a t the point s divided by its value a t the point p. If the point s is on or below BE, the error can never reach h2 times the quantity assumed to be negligible in fixing the ordinate of E. A p p r o x . izia:-

[ [odd]@’c], = ( h Z / z ) . [ [ o d d ] W ~ ] v

(62)

+ { (h?/z).[Rem:[odd]@’c - Ria: [odd]@’c],- [Rem:[odd]9’c],)

here hZ= ~ * / 4 g * where x~ gx is to be given the value it has in the series at p. If v lies below G and p below BE, i.e. if p lies within 0 JE, the error, which is practically h2.[RIb: [odd]@’c], - [R,,: [odd]@’c],, can never reach (h2/9 I) times the quantity assumed to be negligible in fixing the ordinate of E.

+

+

[[even]@’c], = (hZ/2).[[odd]@’c],.{see below} These are the quantities that arise in evaluating Solutions S o s .

I

(63)

and z of Table I.

2836


here hz=+/4g*x2 where gx is to be given the value it has in the series a t p, and the quantity within { 1 is the same as in ( 6 2 ) except that-[Rem:[odd]Wc], is to be replaced by [Rem:[even]Wc],; so that if p lies below OJE the error cannot exceed that given for (62).

+

Ezample f6:-To evaluate [euen]*’c when f(u)=e-*%, t=o.oz, a = o . a j , g z = n / I o , b * = z . This series is represented on Fig. 2 by the point [.\/al=o.o71, g z = n / r o ] , which lies within OJE well below BE. I n Eq. 63, h* = 2 j , so that the constants of the auxiliary series are: t=o.oz, a=6.25, g z = n / z , P = z ; and by Rule I (p. 2840) the error cannot reach 10-12. To evaluate [[odd]*’c],, the expressions for S and V given in Table XI1 No. 6 must be differentiated with respect to gz as the only variable, and then the values of the constants introduced; theresult is [[oddIS‘],- [[odd]V’l, = --0.1914878 0.1910678 = -0.0004200 which (multiplied by 12. j) gives -0.0052 j o as the sum required; five terms of the auxiliary Evanescent were required. The same series was evaluated by the use of the Original Form (approz. vu); this gives [euen]S’ - [evenli” = -1.256014 1.2j0764 = -o.oojzjo the same as before, but 24 terms of the Evanescent had to be computed.

+

+

[Approx. iva]:Approx. i v to cP.c was based on the computation of R,,:@c from the difference between @c and P:+c when gx = 0 ; since W c = P : W c when gx = 0 ,it is not possible to make use of an auxiliary series a t w in computing Wc for the point p. Thus if Table X is used in choosing an approximation t o Wc, approx. iiia should be used where the table says approx. iv. Approx. vu, the Original Form:9‘c = S f - V ’ ; these are found by differentiating with respect to gx the expressions for S and V given in Table XII. Appi,ox. via, viia, viiia:-The procedure is the same as in approx. vi, vii and v i i i , but in Eqs. 5 7 and 58, @c, 8 c , P : 8 c and Rem:@ must be replaced by Wc, e’c, A:8’cand Rem:B’c respectively, and cos S g x o n the right of (j8) must be replaced by - hr.sin Ngx. Approx. ixa:-The procedure is the same as in approx. ix, except that values of 8’c are to be used instead of values of Bc. Sec. 10: Evaluation of the Stationary Series’ of the GFamily’ (a) Exact finite expressions for the sum of S:QE when f(u) is of Class A, B or C In these three Classes, f(u)= up.eBuwhere B may be real, imaginary, complex or zero, and p is a positive integer or zero; with this f(u), Poisson’s Formula gives the exact value of S :@ E in finite form. hnumber of these exact expressions and their P-approximations are given in Table XII; all can be deduced from the expression for S : [even]@cwhere f(u) = ebSuand b is real, by differentiation integration or appropriate substitutions. ( I ) From Eq. 16, setting f(u) = S:[even]*c =

eb2‘l,

there follow

. cos zngr)/(4n*a+b”)

eb21

The Poisson Form3 can be found by setting f ( n w ) = (cos 2ng2)/(4n2a+b2) and Eq. 2 j e of Table VII. Thus

(65) w =I,

in

For definitions and symbols, see Eq. 16 and 18, p. 2803. Stationary Series of the *-Family, see p. 2804 and p. 2856. The definite integrals involved are given by de Haan, loc. cit., Table 160 KO.5 and Table 163 No. I .

* For


2837

S: [even]@c= ( ~ / 4 b G , )eb2[.ebbQZrVa - ebzt/zb9 + ( T / 4 b G ) , e b 2 t . ~ ( ~ - b ( n x - Q z ) , ' &+ e - b ( n s + o r ) / d / ; )

(66)

Summing the geometrical series and rearranging gives (see Table XII, No. 4) T . eb". cosh b ( a - z g x ) / & S:[even]@c= 4b4a sinh b T / f i

-ehZf 2b2

The two items on the right of the first line of (66) constitute the P-approximation ( P : S ) and the second line gives the Remainder (Rem:S)when f(u) = ebZuand @ E = [euen]Qc. Expressions correflponding to Eqs. 66 and 67 when the type of summation is [all!, [ o d d ] , etc., may be obtained as shewn in the "Summary" p. 2801,and in the note to Table XII. ( 2 ) If b in Eq. 67 be replaced by ib where i = 42,there results U(u)=e-*%]

T.e-b2L cos b ( T - z n g x ) / d G S :[even]Qc = - . 4b4a

sin b n - / d z i

+-e-'''b? 2

(Table XI1 So. 6)

The expressions for P:S and Rem:S when f(u) = e-bzu contain imaginary terms which cancel in the expression for S itself, and the real part of P:S is not an approximation to the value of S xhen g --* 0. ( 3 ) The expressions given in Table XI1 No. 5 for S and P:S when f(u) = wehzu can be obtained from Eqs. 66 and 67 by differentiating with respect to b2; those whenf(uj = u?,eb'", by differentiating twice, etc. When f(u)= u p , P:S and Rem:S are infinite, but expressions for S may be obtained by taking the limits as b o of the expressions found for S \vhen,f(u) = uP.ebc; see Table XI1 Nos. 1-3. (4) If b2 in Eq. 67 be replaced by ib2 and -ib2 successively, half the sum of the resulting expressions givesSwhenf(u) = cos b2u (Table XI1 No. 9) while their difference divided by zi gives S when f(u)= sin b*u (Table XI1 No. 8). The expressions given in Table XI1 Sos. I O and X I when f(u)= eK".sin or cos b?u, may be obtained in a similar manner by substituting K+ib2 and K -ib? for b? in Eq. 67. Expressions for P:S and R e m : S may b t obtained in like manner from Eq. 66; with all four forms of f(u)the expressions for P:S are real, and when g -+ o approach those for S.

-

(lob) The P-approximation to S:@E when f(u) is of Class B or C The use of P:S as an approximation to S,when permissible, is advantageous because of the relatively simple form of P:S; but as has been shewn in the preceding paragraphs, real and finit,e expressions for P:S are known only for certain forms of f(u),viz. K o s . 4, 5 and 8-11of Table XII. Since this approximation consists in effect in replacing sinh y or cosh y by e y / 2 , where y stands for the quantities bracketed in the definitions of fi, f 2 , etc. given a t foot of Table S I I a , it follows that the larger these quantities are, the less will be the error of the approximation. Comparison of the numerical yalue of the sinh or cosh with that of the exponential, usually gives a close enough estimate of the error introduced by the approximation (see Ex. 4, p. 2828); but since Rem:S can be evaluated whenever P:S can be evaluated, it is always possible to compute this error in any doubtful case.

S :@.E by means of semi-convergent series' when f(u) is of Class D or E. Whenf(u) has any of the forms listed in Table XI11 or under Class D or E in Table X, each term of the series S : @ c may be replaced by a semi\ There are cases in which it is easier to compute the value of S from S = \' + ac; but (1Oc) Evaluation of

this procedure is obviously excluded when S is wanted only in order to find @c.

W . LASH MILLER AND A. R . GORDON

2838

convergent power-series which for large values of iV may be used to compute the value of that term. To compute S : Cpc, the terms with small values of 1V are evaluated individually, and by means of the power series an expression is found for the sum of all the others (Ex. 1 7 ) . The values of S:@cso found may be used in computing the value of Cpc itself, but this is not recommended, see Ex. IO, 1 2 and 13. Ezampld P7:-To evaluate S:[even]+c whenf(u) = I/-, t = 1 . 4 4 , a = o . z g , gz= b2=o.36. The expreeaion for S may be found from that for the Original Integral entered in Table XI11 No. 2 , by setting u = t and N =zn, multiplying by cos angz, and preBy Table fixing the sign of summation; this gives s=Zcos z n g z . F ( z n m ) / n . \ / - . XXV No. 3 , when yL IO, F(y) may be replaced by 1 / 2 y I/4y3 3/8y5 without introducing an error greater than one unit in the seventh decimal place; hence if c* be written for 4ab2+4at = 1.80, F ( n c ) may be replaced by the power series when n B 8; therefoie 2n/5,

+

+

a

7

s - + z c o s z n g x . F ( n c ) / n ~ + ( ~ / z c ~ ) . z (2ngz)/n2 cos n =I

n =8

The seven terms of the fmt series on the right were computed and added, using the values of F given in Table XXVIII. The second, third and fourth series on the right might have been evaluated by means of Eq. 2 5 d , but except when g z = o or is very small, the labour of evaluation by this method is prohibitive because of the slow convergence of the successive 7

W

derivatives; their values were found by means of the relation

= n =8

-

since exact n =I

expressions for the three sums from n = I to n = co are given in Table XIIb. The result is S + -0.6953506 0.0002880 0.0000029 o.oooooo~ = -0.695060, exact to the sixth decimal place.

+

+

+

(10d) Evaluation of S :@F and related series [The Stationary Series arising from series of the +F Group differ from those from the + E Group only in the occurrence of sin N X in place of cos N X and vice versa. When f(u) is of the form uP.eBv exact finite expressions can be found for the sum of S:*F from those for Z(-nX)/(n+b), which in turn can be secured by expanding Z(n+b) X, introducing the values of &os n X and Zsin n X from Table VI, integrating between the limits II and X,and eliminating. But the expreeaions so obtained, though finite, involve the integrals of bz . cot 2 / 2 d x which cannot be reduced to tabulated forms.’ The P-approximation to Z(sin n X ) / ( n z - b Z ) can be expressed in terms of the Sine and Cosine Integrals, and an approximate expression can be found for the Remainder by using the expansion of Eq. 25d Table VII; but a t best the result is clumsy, and if in it b be replaced by i b (to sum Z ( s i n n X ) / m 2 ) the resulting integrals are untabulated. The sum of Zsin n X (Table XVIII No. I ) has already been found (p. 2802); by integration with respect to X between the limits r a n d X , that of Z(ms n X ) / n i.e. S:[aZZ]Q.‘swhenf(u) = I , can be obtained (Table XVIII No. z ) , but repetition of this procedure is useless because there are no tables of therequisite integrals. Finally although Z(ms n X ) / n 3and Z ( s i n n X ) / n2can be evaluated as shewn in Ex. 18 and 19, the method there employed does not lend itself to generalization. In the special caee that X = o , and f ( u ) = ~ - u p . e * * ~the , sums of the series S:Ws (vis. those obtained from S:+s by differentiation with respect to X = g z ) can he expreased

-

.

Legendre [“Ezer+:ees de &uZ intdagrd,” Vol. 2 , p. 163, (181i).]shews how one of these series,viz. Z ( - i ) n . (an n X ) / ( ? + b ) can be expressed in terms of the derivatives of log r; but his formulas are not convement for purposes of computation.


2839

in finite form by means of the logarithmic derivate of the Gamma-Function; one of these (Table XVIII No. 7 ) is required for use in Ex. 25, p. 2876. The relation entered as No. 3 of Table XVIII was established by Legendre, the inventor of the Gamma-Function. No. 4 can be deduced from Table VI1 Eq. 25d by setting P =o, Q = co,w = I , f(nw) = - I/(n+b) and adding ZI/n to the expression so obtained’ (Legendre arrived a t it through Euler’s Integrals). No. 5 follows from No. 3 and the expression for ZI/(nZ-b*) given in Table X I I a No. 2; No. 6 from Nos. 3 and 5 by addition. No. 8 may be obtained by substituting in turn i b and -ib for b in No. 4, and subtracting; the expression so obtained is convenient when b is large, for small values of b it is better to sum a few terms of the series directly and use Table VI1 Eq. agd to find an expression for the rest of them. The series of Table XVIII with the exception of No. 7 are of the [all]type; the sums of series of the other types can be found as explained in the “Summary,” p. 2801.

TABLE XVIII * 2.ZsinnX 2. Zcos n X / n 3 . b.ZI/n(n+b)

= cot ( X / 2 ) = - log sin (X/2) - log 2. = C ~ / b $(b) = C log b I / 2 b - B1/2b2 B3/4b4- Bs/6b6+. 4. 5 . -b.Zr/n(n-b) = C a.cot bn $(b) I/b n.cot bn 2.$(b) 6. -2b2ZI/n(n2-b2) = 2C z.$(b) - $ ( b / 2 ) 7 . -2b2ZI/m(m2-b2)= C - (s/2).tnn ba/z 8. b2.ZI/n(n2+b2) = C log b B1/2b2 B3/4b4 Bs/6b6+ . . *Here C stands for Euler’s Constant,vie. 0,5772 I 566 49015, B, B3etc.forthe Bernoulli coefficients (see p. 2806), and $(b) for d.logr(b)/db. Tables of +(b) from b = I to b = I O I at intervals of 0.5 are iven in the Brit. Ass. Report for 1916, p. 12j-6; when b lies between I and 2 , values of +(by can be found by tabular differentiation of tables of log r ( b ) , as in Example 25, p. 2876; when b = a p (where a lies between I and 2 and p is a positive inte er) the reduction formula +(a p) = $(a p - I ) ~ / ( a p - I ) must be used. #or purposes of differentiation, the Ia-figure table of Legendre (loc. n‘t. Vol. 2 , pg. 8jand fist second and third differences 95) is best, since the tabular interval is only 0.001, are printed; his logarithms however are to the base IO, in order to arrive a t $(b) the result must be multiplied by 2.3025 8509 29940. Probably the most accessible table is the Brit. Ass. “Table of the logarithmic Gamma-Function’’ (1916Report, pg. 123),natural logarithms, I O figures, a t intervals of 0.005. I.

+ + + + + + +

Ezample 18:-To

+ +

+ +

+

+

+

+ + +

.

+

+

+

evaluate C(cos nB)/n3. Iff(u) = G, t

-

To obtain the S and V of (68), 2: must be prefixed to the integral of Table XI11 No. 4 after writing 2n for N and multiplying by cos zngz; u must then be replaced by t to give S and by zero to give Ti. By setting 2gz = B and giving b* the special value bZ = t in the expressionsz so obtained (and also in the two integrals on the right of Eq. 68) there results, after rearranging and multiplying through by 1 6 a G ,

-.

+ ( 1 6 a G ) . ( R e m : [ e v e n ] Y cwithf(u) = fi and g s = B/2}

1

ZI/n

-

p/n

=

C

=

Euler’s Constant; see Table XVIII.

*This substitution converts the two series into S : Y c and V:Pcrespectively, and the expression on the right of (68) into P:.kc, in all of whichf(u) = dii;see p. 2856. Ei(-2) = (e-Y/y) dy, tabulated by J. W.L. Glaisher: Phil. Trans., 160,36; (1875), and by 1%‘. Lash Miller and T. R. Rosebrugh: Trans; Roy. 8oc. Can.

L’

.

2840

W . LASH MILLER AND A. R. GORDON

Since the left of (69) is independent of at, so must the right be; it is therefore allowable

to select a value of at smallenough to make the last lineof (69) negligible, and yet largeenough to permit ready evaluation of the two series. As the relations of Fig. 2 hold for * e as well as for @c, the best value of at is given by the intersection of B E with the ordinategz = B / 2 . If, for example’, B = r/4,and the value 4;~ = 0.4 be chosen, the quantity neglected cannot Jilt where f = o . 1 6 / a and M = 0 . 4 / & Eq. 69 reach 0.6x1o-6, i.e. ( 1 6 a G ) 10-0 with this value of fi, gives 0.663591 as the value of the serieswhen I3 = n/4;only four terms from each of the two series on the right of (69) were required. Ezample 19:-To evaluate Z(sin n B ) / n 2 . If both sides of (69) be differentiated with respect to B, the resulting equation will serve; if, for instance, B = r / 4 and at = 0.16 ashefore, the result is Z(sin nB)/n* = 0.98188, exact within a unit of the fifth decimal place, since the quantity neglected is less than ( 1 6 a G ) X8 X IO-^. M t = 0.5 X IO-^, see p. 2835.1

.

.

Sec. 11 : The accuracy of approximations to (PE based on Poisson’s Formula; the @E Remainder* Discussion of methods for finding an upper limit to Rem: @ E is necessary because all the approximations of See. 9 except u and iz involve neglecting a t least one Remainder; and as has been pointed out already, the integrals occurring in Rem.@E are untabulxted except for series of Class A. It is of course always possible to evaluate Rem : @ E by graphical integration or by the method of tabular integration illustrated in approz. iz.,and because of the rapid decrease of with increase in u, this does not involve as much labour as might be expected; but in practice the use of Rules which give an upper limit to the absolute value of Rem:@ E is all that is necessary. A very convenient Rule of this kind has been deduced in See. 8:-If the point representing the series in a diagram such as Fig. 2 lies below the line BE, the error introduced by neglecting Rem: @ E will be less than that specified by the diagram. If the point should lie a little above BE, Rules I, IV and V may be used to find an upper limit to its value. The remaining Rules shew for what forms of f(u)and for what values of g or t , the upper limit computed for another f(u)t and g may be relied on. Three of the Rules viz. I, I1 and 111, hold for all forms of f(u);the others are restricted, Rule VI most of all. Rules V and VI deal with relative errors, but in certain cases furnish upper limits to the absolute errors as well. Rule I : The M a z f ( u ) Rule; valid for all forms off(u). If M be the greatest absolute value attained by f(u) while u increases from o to t , it follows from the definitions given in Eqs. 44 4j and 47, that theabsolute3value of R,: @Emust be less than it would be iff(u) were replaced For B = 0,the series has long ago been evaluated by means of Eq. 25d; its value is 1.2020569. * I n the Rules enumerated in this Section, may be substituted for @ (for the definition of [alZ]*c, see Table I11 No. 9) if the following changes are made-Rules I-IIIa stand as rinted, except that therestriction on the values of at in Rules I11 and I I I a must be deleted. f n Rules IV and V, the subscript u = o in Eqs. 73 and 74 must be replaced by u = t. Rule VI becomes “ R n / A for the series involving fi(u) will he greater than R n / A for the series involving f%(u),”so that the order in which the functions are arranged in the following paragraphs of the Rule must be reversed. a The absolute value, Le. the numerical value taken with the sign, will be indicated in the usual manner by two perpendiculars.

*

+

2841


byM;orinsymbols IRn:@E[ z a ( t - u ) , i.e. if gz = n j 2 , at < +/a; if gz = 0,at < + / z . Similar differentiations shea. that the limits just found for RI, will also serve safely for RIb, Ria, etc. The derivative of Rem:[odd]Qc( I ) with respect to t will be positive provided at < 0.3 if gz = n / z , and for higher values of at if gz < X / Z , up to at = I . ? when gz = 0 ;because for these values of at, the absolute value of the increment of each term of the Remainder for a given increment of 1 is less than that of the preceding term.


2843

Thus if the current does not change its direction in the electrolytic experiments referred to under Rule 111, the absolute values of R,: @ E and Rem: computed for any instant of the electrolysis will be less than those for any later instant; whereas if the current does change direction (sinusoidal current for example) all that can be said is, that the upper limits found by Rule I will be less (see Rule 111). In Ex. 2 for example, 2 4 X IO^ is an upper limit to the error when t 6 0.09; but the true value of that error, viz. 8 x IO-^, is not necessarily so.

Rule I V: The Max. Exponential Rule; valid only if f(u) retains its sign i n the interval o < u < t. It is obvious from the equations of Tab. VI11 that for given values of the will be greatest when u = 0 ; therefore, constants at and gx, d/t-u.R,:B~ provided f(u) retains its sign in the interval o < u < t,

In the case of [evenlac, Rem may be substituted for R, in the statement of this Rule; while in the case of [ o d d l a c, Rem may be substituted for R, on the left of (73) and R1 for Rn on the right. This Rule is not of much practical use, since the upper limit it predicts is usually much higher than that given by Rule I ; it can be useful only if f ( u ) decreases very rapidly with increase in u ; for example, if a = I gx = o t = 0.05 and f(u)=e--50u, Rem:[odd]@c is less than o.oo02 by Rule I\', and less than 0.0007 by Rule I.

Rule V: The R,/A Rule; valid only i f f(u)retains its sign in the interval o < u < t. Since for given values of the constants a t and gx, the ratio ( R , : e E ) / ( A :0,) has its greatest' value when u = 0, it follows that (provided f ( u ) retains its sign in the interval o < u < t )

II

as both sides of (74) are positive, the are omitted. In the case of [even]@c, Rem may be substituted for R, on both sides of (74); while in the case of [odd]@c,Rem may be substituted on the left and R1 on the right. Rule V which provides an upper limit to the quotient Rem:@ E by A :@ E obviously provides an upper limit to Rem:@E itself whenever the value of A: can be found. The quantity on the right of (74) usually gives a rather high upper limit to the quantity on the left, especially when gx is small; but on the other hand, this limit is independent of the form of f(u)and is easy to compute. For example, no matter what f(u)may be (so long as it retains its sign), if t = 7 / 9 0 , a = I , and gx = a/3,the quotient Rem: [oddlacby A : [odd]@c must (by Eq. 74) be less than 2.6 X IO+; from Tab. XXVII, when f(u)= I the true value of this quotient is 8 X IO-^. This is true if the series be of the [even] or [odd] type and gz lies between o and s/z;

see Eqs. 27-32 p. 2809.

2844


If the series be of the [odd] type, the percentage error introduced by neglecting R e m : [odd]@c is almost exactly one hundred times the quotient ( R e m : [ o d d ] + c ) / ( A :[odd]%c); and Eq. 7 4 states that this percentage error cannot exceed 200 times e - - n ’ ’ 4 a t . ~ ~ (~nhg z / z a t ) , no matter what f(u)may be so long as it has the same sign between u = o and u = t . For an application of this Rule, see p. 2 8 5 5 , also approx. c, p. 2960, and “Boundaries for Eqs. 1 1 0 - I X Z , ”p. 2869. Rule VZ: T h e d . log f(u) Rule; valid only i f f(u) retains its sign in the interval o < u < t and steadily increases or decreases during that interval. If two series of the same type of summation and with the same values of the constants, a t and gz, differ only in respect tof(u) (assumed positive’ over the whole range of integration in both cases), and if for every value of u between o and t, d log fl(u)/du is greater than d logf2(u)/du1then R,: @ E / A : for the series involving f ~ ( u )will be less2 than R n : @ E / A : Qfor ~ the series involving fz(u). From this Rule i t is obvious that R , / A when f(u)= e*% will decrease as b increases, while forf(u) = e-”%, the opposite is true; also that R , / A is less for f(u) = b2+u than for f(u) = .\/b2+U;less for f(u) = b2+u than for f(u) = zb2+u, etc. Of course the type of summation in the series compared, and the values of a t and gx, must be the same. When the value of, or an upper limit to, R , / A has been found for some particular f(u),Rule VI defines a group of forms of f(u) for which this is an c ) f(u) = I , upper limit. Thus the value of R ~ : [ o d d ] @ c ) / ( A : [ o d d ] @when a = I, t = 7/90 and gr = n/3, vie. 8 X IO@, (or the upper limit to it found by Rule V, vie. 2.6 X IO+), constitutes an upper limit for every series [odd]& when f(u) is such that d log f(u)/du is positive and the constants a t and gx have the values given above; such functions are:-f(u) = u, u2, b2+u, e*%, etc. Thus the error introduced by substituting P : [odd]@cfor [oddlac cannot exceed 8 x IO-^ percent of the sum of the series if f(u)= eu, and will exceed 8 x IO-^ percent if f(u)= e-U; but is obvious that the absolute error will be greater in the former case than in the latter. This assumption does not involve any loss in generality; for so long as f ( u j retains its sign, changing this sign does not affect the absolute value of the sum of the series. 2 This Rule may be established by the following argument:-Writing A as an abbreviation for A : @ ~ a nA‘for d thequantity ( A : B ~ j f ( u )it, isobvious that thequantity (A’/A).du must always be a positive proper fraction no matter what f(u)may be; the sum of all these fractions is obviously unity; and the sum of the products of each one of them into the corresponding value of the ratio R ~ : ~ E / A : B(via. E that taken for the same value of u as is the fraction) gives ( R n : 9 ~ ) / A .If d Zoqf,(u)/du be greater than d ZogfZ(uj/du for every u between o and t , there must be some value of u in the range at which the two fractions are equal; at all higher values of u, every fraction for f i ( u ) will be greater than the corresponding fraction for f2(u), and vice versa. Now the values of the ratio Rn:BE/A:Bs into which these fractions are multiplied are independent of f(u),i.e. they are the same for the two sets of fractions; moreover (see Rule V) the values of Rn: BE/A: BE decrease as u increases. Therefore the series involving f ~ ( u )which gives rise t o the greater fractions a t the lower values of u will have the greater Rnl.4.


PART IV EVALUATION OF SERIES OF THE AB GROUP INCLUDING THOSE OF TAB. I1 The general definition of the A-Family is given on p. 2 7 9 5 , see also Tab.

I11 KO.I O ; the AE Group is defined on p. 2800; the Poisson Form and the definition of P : A and Rem:A are given in Eq. 48, p. 2 8 1 2 . The Solutions of Table I1 (see Tab. IV) each consist of the sum or difference of two A’s in one of which X = g(z+X), and in the other X=g(z-A), see Tab. I11 and VIII. The Original F o r m (approx.ub):- As at remains constant during the Xintegration, there is nothing in this Family to correspond to the Stationary Series of Part 111. Every series of the A-Family contains a converging exponential, and the “number of terms of V” in Tab. XV gives an upper limit’ to the “number of terms of [even or odd]Ac.” The Poisson Form:- This is more generally useful for purposes of computation than is the case with series of the @-Family, because the integrals involved have been tabulated not only when F ( X ) = X p ( p a positive integer), but also when F ( X ) = eBXZfCX if B and C are real. The untabulated forms which arise when B or C is imaginary or complex are dealt with in the Appendix, p. 2 8 6 2 (see Table XXI p. 2865). Boundaries:- If F ( X ) = I, a safe Boundary for Rem:lcanalogous to B E of Fig. 2 can be found from

the upper sign to be taken if X = g(z+X) and the lower if X = g(z-A). The quantity within brackets on the right of (75) is the difference between two Probability Integrals, viz. that in which the argument is U and that in which it is L, where U = (n-gz)/ad/at and L = (r/z-gz)/z.\/zif X = g(z+X), and where II = (3ir/z-gz)/24/at and L = ( n - g 2 ) / 2 & if X = g(z-X). Safe upper limits to the values of R,,:iic are found by setting U = 30 in Eq. 75. If X = g(s-X) these upper limits depend only on the values of 1 and of ( r - g z ) / d z , so that the upper limit of (RIa)/L has the same value for all series represented by points on any line passing through Q of Fig. 2 ;the highest line for which this upper limit is negligible constitutes the Boundary for R,, and also for Rem. Table X I X gives the values of .\/at for the point a t which the Boundary2 cuts PH, when the “permissible error” (i.e. the greatest quantity assumed to be negligible) varies from 10-8.1 to IO-~.Z. If X = g(z+A) the Boundary for R,, runs upward through P; the ordinates of its intersection with OC are also given by the values of -\/atin Table XIX. If the Scales of Fig. 2 are used in estimating the number of terms of Ac, the error will not exceed IO-~..,W where M is the maximum absolute value attained by F ( A ) in the interval o < A < 1. The line BE of Fig. 2 (in which E = 0.227) is the Boundary for Rem:Ac (with X = gz - gX) which implies an accuracy of 6 X ~o-’.Ml, see Table XIX. The correspon_ding Boundary when X = gz gX would be constituted by a line from P to meet OB a t v‘d =

+

0.227.

2846


TABLE XIX Boundary for Ria: [even or oddlhc when F(X) = I . Ordinates on OC for the Boundary when X = g(x+X); ordinates on P H for the Boundary when X = g(x-X). (Ria: [even or odd]iic)/l 10-8 10-7 IO-^ 10-5 10-1 10-3

dZ

0.197

0.212

0.232

0.260

0.299 0.359

A p p r o x . ib, the P-upproximution;- If the point representing the series lies below the Boundary, the error introduced by substituting P:Ac for h c cannot exceed M times the quantity assumed to be negligible in fixing the ordinate( on OC or P H as the case may be) for F(X) = I ; M being the maximum absolute value attained by F(X) in the interval o < X < 1. AS the integral in A :Ac has the same form for all values of gx, the simplification introduced by setting gx = o in series of the @E Group has no counterpart here. Approx. ii, iii and iv of Part I11 cannot be adapted, because in this Family there are no relations corresponding to those developed in paragraphs ( a ) to ( d ) of Sec. 8. A p p r o x . vi-viii of Part I11 are not applicable because in this Family the integration is with respect to the angle, whereas in the @-Family it affects the exponential on which the convergence of the Original Form depends. A p p r o z . ixb;- In this Family the quantity to be tabulated is Bc.F(X); the values of 8 c in such a table are all for the same value of T = at and successive values of X = g(z A),

Sec. 12: Evaluation of Multiple Series occurring in the 1 1 Group ~ D o u b l e Series;- It has been shewn in Tab. IV that the series printed a t the head of Tab. I1 can always be expressed as the sum or difference of two series of the b E Group, in one of which X = g(x+X) and in the other X = g(x-X). If now the F(X) of these series be one of the series occurring in the Solutions of Tab. I (in which 9 must be written for t and X for x) i.e. if F(X) be [even or odd]@cor W c , the A-series so arising will contain a double summation; approximations to the values of these Double Series will now be considered. Taking F(X) = [even or odd]@cas an example,

in which Q stands for 24 or 24 - I and S for zn or zn - I ; since the two summations are independent it is necessary to use different symbols for the two variables of summation, but because of the brackets there is no ambiguity W

in writing Z for

in the first and Z for *=I

E

in the second. The double sign

n=I

is used in cos Qg(x=ttX)to include both the Ac’s; and where the double sign occurs in Eqs. 77-83, the upper is to be taken if has been taken in (76).

+


2847

Since in Eq. 76 Ac is presented as the product of two series, in one of which q is the only variable of summation and in the other n, its value must be independent of the order' in which the summations are effected. As moreover each of the quantities a 29- gx and I is independent of all the others and of n and q, the four operations (viz. two summations and two integrations) indicated in Eq. 7 6 , and also any differentiations with respect to a 29- g s or 1, may be performed in any order that may prove convenient; and a t any stage either or both sums may be replaced by any suitable approximation. I n practice it is best to begin with the integration with respect to u, consulting Table XI1 and writing h wherever s is printed in that Table. This replaces the second factor in the h integrand of (76) by S - B = S - Z cos h'gA e-''20B func N (see Eq. 1 6 ) , and thus converts the right of ( 7 6 ) into the difference between two series; one of these, if S is known in finite form will only involve a single variable of summation viz. q, while the other is a double series. I n this new double series all the factors of the integrand are free from A (and are therefore to be treated as constants during the integration) except cos Qg(z +h).cos iVgA whose integral between the limits o and 1 is

[ A V = z n ,Q = z q - ~ ] (+Q sin Q g x

+ (-I)"+'.Q

cosQgx)/g(-V2-QZ)

Thus whenever a finite expression is known for the sum of S, i.e. if to Class A, B or C of Tab. X, Eq. 76 can be p u t in the form?

Ac

=Jk'.Ze-Q'"('-".cos

Qg(z&tX).dx - ZZfactor .e-Q2a(' -)'

(77)

belongs

@E

.e-"'"S/g( ,Vz-Qz)

(79)

where the "factor" of the double sum is

(+&.sin Q g x - (-I)"+'.

,V.eos Q g x ) func N

if K is odd and Q is wen, or a similar expression in the other case. The single series in Eq. 7 9 will be rapidly convergent if a ( t - 9 ) is large, and in such cases may be summed term by term; the integration of S cos Qg(z&h) presents no difficulty. If a(t--rl)) be small, the series (which is a 0,) may be replaced by its P-approximation from Table VIII; the subsequent This is important, for there are many repeated series whose values depend on the order of summation; for example Z Z I / ( ~-~4q2) in which m stands for a n - I and both n and q must be given all integral values from I to m . I n such cases the order in which the summations are to be carried out is indicated by the relative position of the signs of summation, thus B 2 indicates that the value of B 1/(m2-4qZ) is to be found for q = I , z , . ., and

.

n

9 %

that the sum of all these values is to be taken as the value of the repeated series; since Z ~ / ( r n-~44?) = o for every value of the integer q, it follows that Z Z 1/(m2-4qZ) = 0 , while 2 Z 1 / ( m 2 - 4 p ) n q

=

Z -1/zm2 n

= -n2/16.

Q "

See Bromwich: "Introduction to the

Theory of Infinite Series,'' p. 78-81. * Since the value of ,le is independent of the order in which the summations are performed, the symbol Z Z which leaves the order undetermined is used in Eq. 79.

2848


integration will lead to tabulated forms only if f(u) = corutani, up or e**%, in other cases tabular integration (see approz. iz) must be employed. The double series in Eq. 79 may be summed term by term if both at9 and a ( t - 8 ) are large; in other cases it should be differentiated with respect to at9, which removes the denominator ( -Y2-Q2); in the derivative via.

the variables are separated, and any or all of the single series may be replaced by suitable approximations.1 iifter the replacements the derivative must be multiplied by d(at9) and integrated with a29 as the upper limit, not forgetting the integration constant. The foregoing is essentially the procedure followed in Illustration I; the double series of (893) however involves two A's as must all that arise from the Solutions of Tab. 11; for reasons of convenience it was combined with still another series to give the double series of (85b) before performing the differentiation and replacements, The series evaluated in this Illustration arises in problems discussed by Rosebrugh and Miller (loe. cit., p. 862); to facilitate cross reference, their symbols have been retained and the [odd-alt]%s of the original has not been replaced by [odd]%c. The method just discussed for dealing with the sum on the right of (79) presumes the existence of a finite expression for S ; it is moreover restricted by the assumption that if N is even Q will be odd and vice versa. The first limitation is unavoidable; the second is of no practical importance, for when N and Q are both odd or both even the physical problem reduces to one in which the Boundary Conditions remain unchanged except for a discontinuity in f ( t ) a t the moment t = 9, and the Solution does not contain a double series. Examination of the integral of cos Qg(z *X).cos XgX when Ar and Q are both of the same type, and in particular of its form when N = Q (which obviously cannot occur if one is odd and the other even), shews that in such cases the double series of Eq. 79 reduces to a single series; but for the purpose of the present paper detailed discussion of this transformation would be superfluous. The difficulties which arise when putting this procedure in operation, are those of the integration with respect to at9; if the n-summation of the double series in (80) be replaced by its approximation while the q-summation is left as it stands, the integration leads to tabulated forms in all cases where a finite expression for S is known except whenf(u) = ek".(sin or cos) b2u;but if the q-summation be replaced (whether the n-sunimation be replaced or not) the integrals are untabulated except for g s = 0 . The form of the integrationconstant must also be taken into consideration; the object of the replacement is to save labour of computation, and when the integration-constant is clumsy much of the advantage is lost. ' T h e q-series are of the form e or e', approximations to whic,h may be found from Table VIII. The n-series are of the form V = S - 9 or V ' = S - *', all for gz = o and thef(u) of Eq. 76; so that approz. i, ii and is (or ia, iia,and iza) are available.

SVMERICAL EVALUATION OF INFINITE SERIES

2849

Triple Series:-If in the Solutions of Table 11, the F ( X ) be replaced by the double series of Eq. 79 (writing tl for t and X for x), the A’s will each contain a Triple Series. I n the cases which occur in practice the type of the new summation will be odd if Q of Eq. 79 is even and vice versa, since both the sines and cosines of ( 7 7 ) and ( 7 8 ) are summed with respect to q ; in all other cases the triple series will reduce to a double series and the physical problem to one whose solution does not involve a triple sum. This leads to a certain symmetry in the triple series with respect to the first and third variables of summation, which accounts for the choice of a special method for finding a “ q - a p p r o ~ . Jin J Illustration 11. As with the double series of Eq. 76, the sum of the triple series is independent of the order in which the summations, integrations, differentiations and replacements are effected. Its form is such that, by introducing two differentiations, the method of Eq. 80 may be employed to separate the variables; this is essentially the procedure followed in Illustration 11, except for the “q-approx.” where the integration constants’ proved to be extremely clumsy. There is another procedure which dispenses with the differentiations, and without separating the variables of summation a t once replaces the q-summation by its approximation. This operation reduces the triple series to a double series in which each term contains functions of n and p (the third variable of summation), and in some cases may lead to a very useful approximation to the sum of the triple series. If P be written for 2 p or 2p-I, the triple series must have the form where the ‘ifador” depends on whether & is even or odd. If the upper sign be taken in Eq. 76, these factors are [Q even]

Qz{ *(*(

[ Q odd]

sin Pyx-cos

- I ) n + q . sin Pyx)

Pyx)

+ N P {( -

I ) ~ + ~ . C O SPyx

- & { =t(-~)’+~.P.sin P~z+&.cosP ~ x ) - & { ( - I “ + ~ . & . Pgx cos +(- ~ ) ~ + * . P . sPiYnX )

If the lower sign in (76) be used, Q2 outside the first bracket must be replaced by - Q 2 , and - & outside the first bracket in the third line must be replaced by +&. Since -V and P are both even or both odd, in some terms of the summation they will be equal; for these terms, (81) becomes ze-N2a(9

+t -ti)

.juneAT. ~ j a c ~ o r . e - Q 2 ~ ( f ~i -~ 9 ~ ) )z /y (z ~ ~

(82)

For the terms in which N is not equal to P, (81) may be converted by the method of partial fractions into

In the “n-approx.” and in the “p-approx.” only one differentiation is involved, and the integration constant is zero.

2 8j o

W. LASH MILLER AND A . R . GORDON

Approximations' to all the forms that can occur in these q summations, except such as can be found immediately from Table VIII, are given in Table XX. Some of them are complicated enough; whether in any given case direct summation of the triple series would be preferable to use of the formulas arrived at by this method, depends entirely on the values of the constants; in the comparatively simple series of Illustration I I , this q-approximation is of real importance. TABLE XS Approximations for use with Eqs. 82 and 83 4 1 / 2 n P - (d/?r/2rn).F(&m)

I.

XEe/De

2.

S( - I)*, Ee/De

3.

ZEolDo

+ 1/21712

+ (-

I)n.r.e-m'T/Am

- (v'.Rj3n).F(.\/m)

--f

4.

s(-

-+

(-

j.

SEe)(De)z

-+

(t/K/4

6.

Z(- r ) q . E e / ( D e ) ?

i

- ( - 1)".x.(2mV'+ 1 ) . e - ~ * ~ / 8 m 1/2m4 ~--

7.

ZEO/(DO)'

-+

I ) ~ . ( z q - I).EO/DO

I)n.r.e-qn'T/q

+

~23). ( 2m?T I ) . F ( &T) - dx/4?n2 ~ ~ . e - ~ ' ~ / 1 1/2m4 6 ~ 7 ~ ~ -

+

1.

(d/?r/3 2 n3).( 8 n2T+ I F ( dm) - .\/Z/16n2 ~~.e-~'"~/63n~

+

8. Z ( - I ) * . ( ~ - I ) . E O / ( D O ) ~-+ (-I)'L.TT.e-Jn'T/q m

In this table the following abbreviations are used: 2 = 2 ; Ee = g =I

Eo

=

e-(zq-r)'T; De = @-rn?;

andm =

Do

= ( z q - ~ ) ? - ~ n ?n;

is a positive integer

2n-I.

Illustration I : Double Summation

(a) Origin of the Double Series In the first problem dealt with by Rosebrugh and Miller in Part I1 of their paper, the Initial Condition is [at t = 01 2-20 = 0 ; the Boundary Condition [at z = o ] is z - z a = o for all values of t ; the Boundary Condition [at r = l ] changes at the moment t = 8 from dz,'dl: = CI t o 2 - 2 0 = r - 2 0 ; 20, CI and are constants. Thus for t 48, an expression for 2-20 is given by Table I S o . j a , after replacing z by z - z 0 andf(t) by C1. In particular

8 ] z - z 0 = S u z 0 . z /-lz ( f T Q / n ) 2. ( - 1 ) q + * . s i n

+

(211)

~q~r.e-4@"("--9)/4

. J i s i n zpgz.sin 2 q g X . e -4*20(i -*).F(x) . A

@Sa)

Instead of evaluating the single series of (8 ja) by the methods of Sec. 9, and dealing with the double series as described on p. 2848, it is shorter in this case to convert the former into a double series,' add it to the latter, and deal with both at once; the compact form so obtained, Eq. 8 j b , is identical with Rosebrugh and Miller's Eq. 7 I.

If both sides of ( 8 5 b ) be differentiated with respect to gz, and then z be replaced by 1, there results [at z=1]

a2/ax =

( { - z o ) / l - ( 1 6 C ~ / n ~ ) . ~ B e - " ' " ~ . e - " ~ " /(m?-44?) '~-~) (86)

Eq. 86 occurs in the solutions of certain problems of electrochemical polarization now being studied in this laboratory; an expression free from the restriction z= 1 may be obtained by multiplying the quantity to the right of the sign of double summation by (- I ) ~ " . C O S q g z .

(b) Evaluation of the Double Series If a29 and a ( t - 8 ) are not too small, the double series of (86) may be evaluated by direct summation. But the smaller a 8 and a ( t - 8 ) are, the greater the number of terms that must be computed and added; in such cases the methods of approximation already described may be employed. For brevity the letter H is written for the double series of (86) omitting its coefficient (- 16Cl/s2), A is written for a 8 and E for at. Differentiating with respect to A , aH/aA - ( Z ~ - ~ * " ) . ( B ~ - ~ Y " E - -) ) (87) Either or both of the factors on the right of (87) may now be replaced from Table VI11 by its P-approximation, and if the result be multiplied by d A and integrated, it will give rise to an approximation to H . Three cases are possible:By replacing

< -20

by the expression obtained when x is set equal to 1 in Eq. 84b. 0

2;

In Eq. 84, Z stands for 2 a3 usual, in (8p)for in (85b) both summations are from n=I q=r to m and the order of summation is immaterial.

I

W. L A S H MILLER A S D A. R. G O R D O N

2852

q-approximation:-u(t-

8) small and a 8 large.

the q-summation on the right of (87) being replaced from Table YIII; from Table XIIa No. 2 , the first double series on the left of (88) is equal to - Z e - m ' E / z m 2 . Therefore

H

-+

( d \ / 9 / 2 ) . s e - m ' " . F ( m ~ ~)/m Ze-m'A / 2 n 1 2

n-approximation:-ab

(89)

small, a(t- 9 ) large.

the n-summation on the right of (87) being replaced from Table T'III. From Table XIIa No. 2 , the second double series on the left of (90) is zero. Hence

H

-+

( d G / 4 ) . Z e - 4 g " E - A ).F(%?d/A)/q

Double approximation:-both

H-

zc

(91)

a b and u(t- 8) small.

e -4gZE/(rn2 - 4q2) -+ (a/16).ljd/dd(g-A) ( d P / 8 ) . J0 j A l d x

(92)

The second double series on the left of (92) is zero (Table XIIa KO.2 ) ; hence

H

-+ d * A / 4

Example 20:- a b = A =

0.25,

- (~/8)..~iii-'d/A/E

(93)

ut = E = 0.34.

Direct summation H=-0.183 2412 = -0.183 2413 q-approximation n-approximation = -0.183 243 Double approximation = -0.1831 The approximations are not improved by computing more terms.

(18 terms needed) ( 8 terms needed) ( 6 terms needed) (no summation.)

Illustration I1 : Triple Summation (a) Origin of the Triple Series If in the case dealt with as Illustration Z, at some moment I1 > L p the Boundary Conditions or one of them were to change again, the Solution would contain a triple series. The Boundary Conditions chosen for Illustration ZZ are those of a problem in electrochemical diffusion now being studied in this laboratory; they are:- [at t = o ] t - z 0 = o ; [at x = o ] Z - Z O = O ; [at r = l ]dz/dz=C, for t < b , z = ( for b < t < t l , az/dx=C', for t > t l ; CI Cp and { are constants. When t > t l the solution is the sum of Solutions Table I KO.3a and Table I1 No. z after replacing' the m of Table I1 by r = ~ P - I , t by t-h, f ( t ) by 'These are the letters used by R. and M.;r = 2p - I and D = p of Eq. 81.

NUMERICAL EVALUATION OF INFINITE GERIES

2853

C2, and F(A) by the expression on the right of (85b) in which tl is to be written for t and A for z. Performing the integrations and collecting like terms, Eq. 94 follows immediately; it is identical with Eq. 7 2 of the paper by Rosebrugh and Miller.

[ t > t l ] 2-20

=

czz + P812 -'

{* l z o -

cos mg ( ~ - z ) . e - m ' ~ ( ~ - ~ l ) / m *

___

(b) Evaluation of the Triple Series The triple series on the right of (94) may be evaluated by direct summation if all three quantities ab, a(tl-6) and a(t-tl) are large; as in Illustration I , the order of summation does not affect the result. The seven different approximations from which a choice is to be made when one or more of the three quantities is small are given below, for the special case that z = 1. The letter H is used to indicate the triple series of (94) when z=l, omitting the factor - 128C1E/7r4; A is written for a b , B for at1 and E for at. v-approximation:--a(t-tl) small, the other two large.

The sum of the first series on the left of (9j) is zero (see Table XIIa, S o . 2 ) ; therefore H ---f - (~/.rr/4).BBe-"'A.~-4**(B--A) . F ( 2 d / E - B ) / n ( m 2- 49') (96) n-approximation:-ab small, the other two large. If in (96) r be written for m, r2(E - B ) for m 2 A , z q v ' / A for zqv'\/E-B, the resulting expression will be the n-approximation to H. q-approximation:--a(tl- 8) small, the other two large.

see Eq. 8 2 and 83. Introducing the approximations of Table XX Xos. 5 and I , there results H + z ~ - ~ ~ ( A - B - E ) . { (&/4m3). (zm*B-A+ I) .F(rnd/B-A)++.e --m'(B - A ) 1 16m2 - < T ( B - A)/4m2- I / Z in4J ZZe -rn24.e-r2(E-B) / ( m 2 - r 2 ) . { r/zm?(v'r/zrn). F(rnd/B-A)- 1/2r2 ( & / z r ) . F ( r d / B ) } whenlm*r (97)

+

,

+

Ezample 21:-If A =0.50, B=O.59, E = r . o 9 , two terms of the single series are needed viz. m = I and m = 3; their sum is 0.028921 I . The sum of two terms of the double series (viz. m = ~ r, = 3 and m = 3 , r = I ) is -0.0005958; the symmetry withrespect to vi and r of the last factor of Eq. 97 much facilitates their computation. This gives H=o.o283253; direct summation of the triple series (20 terms) gives 0.0283255. In the double series of (97) and in the triple series of the preceding equation, terms in which m = r are to be omitted.


2854

q-v approximation:-u(t-

t,) and a(tl- 9) small, a 9 large.

Choosing B as upper and A. as lower limit of the integration, this gives -m2B.e - + ( E - B ) E e - m 7 d . e - 4 q z ( B - A ) ,e -r2( E - B )

E

ce

EE

m2- 44’

m2- 4q?

-+

From Table XIIa No. 2 and Table VIII, the sum of the first series on the left ( ~ / d mZ )e - m . Z B / m 2Therefore, . mulof (99) approximates to - (&/8). tiplying through by dB and integrating between the limits B and E .

The sum of the first series on the left of (100) is zero (see Table XIIu, KO.2), and the integral on the right can be evaluated by changing the variable of integration from B to y = m d m , and replacing F(x)by X(I - 2x2/3 4x4/1j . . .), see Table XXV Xo. 2. Therefore

+

+

in which U = nid-

and TV

=

nidE-8.

Example 22:-If A =o.jo, B=o.60 and E=o.69, the n, the n-v, the n-9 and the n-q-u approximations are not appropriate; the value of the series obtained by direct summation, and the values given by the other three approximations are as follow(40 terms needed) H =o.ozg 1760 Direct summation ( I O terms needed) =o.ozg 1760 u-approximation ( I O terms needed) =0.029 1759 q-approximation (19 fractions needed) =0.029 1762 q-u approximation From the point of view of labour involved, there is little to choose between the q-u approximation and direct summation; the other tn-o are much shorter. If however A =0.50, B = o . ~ and I E =o. j 2 , direct summation would be extremely laborious, while the q-u approximation would be much easier to compute than in the example given above.

n-q approximation:--aG and a(tl-21) small, a ( t - t l ) large. If in the expressions on the right of ( 1 0 1 ) r be substituted for 7n, A for E - B, E - B for A , and B for E - A , the resulting expression will be the n-q approximation to H .

NUMERICAL EVALUATIOS OF INFINITE SERIES

n-v approximation:-at9

n-q-v approximation :-all

2855

and a(t-tl) small, a(tl- 9) large.

three small.

Ezainple %?--If A = O . I O , B =o.zo and E = o . 3 0 , any of the seven approximations may be employed; because of the values of the constants, the n-approximation is identical with the u-approximation. Some of the results are: Direct summation n- or v-approximation n-v approximation n-q-v approximation

0.033 0.033 0.033 0.033

8027 8027 8027 794

(57 terms needed) ( 2 2 terms needed) ( 5 terms needed) (no summation)

Errors of the seven preceding approximatio?u:- The error of the n- or qor wtpproximation depends primarily on the value of A , B - A or E - B respectively; a n upper limit may be found by means of Rule T’; the use of Rule I is not so convenient, as in these cases thef(zc) of that Rule appears in the form of a double series. As an illustration, the absolute error of the ?-approximation in Ex. 2 2 cannot exceed 2 H.e-*’ 0 3 6 = z X 0.029 X e--274= 8 X 1 0 - l ~ . An upper limit to the error of the ?I-c approximation can be found in the same way; in Ex. 23 for instance, it cannot exceed 1H.eCK’ O 4o - 1.2 x 10-11, In the case of the q-r approximation, the error introduced by neglecting the z-Remainder is always numerically smaller than that introduced by neglecting the q-Remainder and is of the opposite sign. K h e n Rule V is used to find an upper limit to the latter, the “RIaximum Exponential” of the Rule involves the sum of E - B and B - A , i.e. it is a function of E - A . Thus in Ex. 2 2 the error cannot exceed the product of 4 ( E - - . 4 ) into ’ the expression in square brackets on the right of Eq. 1 0 1 , viz. 6 X 10-7; as shewn in the Example, its actual value is z X 1 0 - 7 . Similarly, in the n-q-t’ approximation the only error which need be considered is that introduced by neglecting the q-Remainder. If Rule \- be used to fix an upper limit to this quantity, the “AIaximum Exponential” is a function of A f B - - d + E - B = E ; thus in Ex. 2 3 , the absolute error of the n-q-u approximation cannot exceed 2 ( r 3 ‘,/16).e-”’ 4 E X (Erpression in { ) on the right of Eq. 103) = j X IO-:; the actual error is 9 x IO&.

W-.LASH MILLER AXD A. R. GORDON

2856

PART V SERIES FOR USE IN EVALUATING CERTAIN FAMILIES OF INTEGRALS In the preceding sections it has been shewn how the sums of certain series may be found by the use of tables of integrals; here it will be shewn how integrals of certain families may be evaluated by the use of series1related to those of the 0-Family-from Tables I11 and VI11 [all]0c = Be-n'T.cos n X = A : [al1]0c - 1 / 2 Rem: [al1]0c ( 104) where A : [a1l]0c = &/2d/?i).e-X2'4T Of the five approximations discussed below, ( a ) is in effect an approximation to the integral of (A.Bc).f(X).dX, ( b ) to the integral of (A:Bc)f(T).dT when X = o , and (c) to the same integral w h e n X i o ; while for approx. ( d ) , Eq. 104 is divided through by A : & and the integral approximated to is that of (1/2).f(X).(1/A:Bc).dX; approx.(e) is in effect an extension of approx.(c). For practical reasons series of the [odd] type are to be preferred in approz.(a), ( b ) and (c); in ( d ) series of the [all] or [even] type must be used. All these approximations are based on series of the 0~ Group; analogous formulas based on series of the OF Groupareof little practical importance because of the slow convergence of the e p Remainder (see p. 2824). The series employed in approz.(b) and (c) are members of the Group of the *-Family, general expressions for which may be obtained by multiplying the corresponding (in which T and X are to be replaced by a u and gz respectively, see Tables I11 and YIII) by f ( n ) . d uand integrating between the limits o and t. In series of the @-Family,f(zr)is by definition independent of t (p. 2 7 9 5 ) ; if f ( t - I L ) be written instead of fiu) the series is converted into one of the *-Family, since

+

P,

J!

Pl

f ( t - u ).du

;-.v~a:l--uJ

=

J! k

-S2nu

.f(u).du

An example of this conversion is given in the foot-note to p. 2839. If f(u)= I , the two series are identical, i.e. *(I) = + ( I ) ; if f(u) = eBu, the relation* is *(eB") = eBt,@(e-B"). A11 that has been said as to the "Boundaries" (p. 2821) and the "number of terms of V" (p. 2822) in connection with series of the @E Group is applicable to series of the * E Group; but the Rules for finding upper limits to Rem: @ E must be modified as stated in the foot-note to p. 2840. The series of approz.(a) are members of the A E Group in which gr = 0 ;the series of approx. ( d ) are suz generzs Sec. 13: Approximations to certain families of integrals Approx. to se-Z'.f(z).dz by series involving s c o s mrf(z).dz. (a) (Originally proposed3 as a general method by W. Thompson in 18jj). From Tables I11 and VIII, choosing [odd]&, In this Part, as throughout, 2 stands for

5, and rn=zn--I. 7l=1

The symbols @( I ) , Q ( e - B u ) etc. are used to indicate series of the @-Familyin which f(u) = ~ , f ( u = ) e--Bu, etc. U'. Thomson: Quar. J. Math., 1, 316 (1857). 2


285 7

Replacing X / z d / T by x, multiplying through by 4 4 / T l n . f ( z ) . d z J integrating between the limits o and K , and rearranging,

/ X z z . f ( z ) . d x = ,J’T.~e”‘‘ 0

iT

JK

cos zrnx&‘.f(x).dz

+ aRemainder

(106)

The approximation consists in neglecting the “Remainder” on the right of (106) ; in order to facilitate the summation, the largest convenient value of T consistent with the accuracy desired should be chosen. To find the error introduced by neglecting the “Remainder,” the latter may be replaced by its first item, which if the maximum absolute value attained by f(z) in the interval o < x < K be represented by M , is less than 13

. 1 1 . ~ K e - ‘ ” - 2 1 ~ ’ 2 , 4 T ) .and d z , this again is less than M.$,

e-u2.dy where L

is written for ( T - z ~ d T ) / 2 & . This last expression becomes identical [even or odd] Ac)/I” of Table X I X if with the product of M& into the “(R,,: the .\/at of that table be replaced by v‘F and the g x by Z K @ ; so that if the quotient, of the “permissible error” by M.\/?r be looked up (interpolated) in the top line of Table XIX, the corresponding figure in the second line will be the ordinate for E in Fig. 2 , i.e. it will fix the position of the Boundary B E ; the point at which the ray (Le. line passing through the origin) with slope I / Z K cuts’ this Boundary gives the desired maximum value of dF, This relation is expressed analytically (writing E for the ordinate of the point Eof Fig. 2 ) by = 2 a E / ( s + q K E ) ; for instance, if K = z and if IO-~.M& be considered negligible (whence . E = 0 . 2 3 2 ) the desired value is d / T = o . 2 9 . If the permissible error be I O - ~ . M & , an upper limit to the number of terms of the series that must be computed is given by the point on the “odd” scale (as printed in Fig. 2 ) whose ordinate is that of the intersection of the ray with BE (or with the perpendicular P E if it does not cut BE); for other assumptions, see Table XV. With F = 0 . 2 3 2 , the number of terms increases with increase in K from a minimum of 4 terms to 9 terms when K’3.35 (where the ray passes through E) after which it mounts rapidly; if K = IO, 30 terms might have to be evaluated and summed to ensure an accuracy of I O - ~ . M ~ G . Thomson suggested that this approximation might be used, setting f(x) = I , to compute the Probability Integral,? but approx.(b) is better. An approximation to the integrals of e-’*.sin or cos hxdx may be found by setting f(z) =sin or cos bx; a? least four terms of the series are needed to ensure an accuracy of I O - ~ . ~ / no X matter how small K may be, and the number increases with increase in K ; when K = 1.3 j seven terms are necessary, and for larger values it is easier to evaluate the integral by computing H(b/2,K ) and K ( b / z ,K ) as shewn in the Appendix p. 2869.

’ If the intersection lies to the right of PE, the ordinate a t which the ray cuts PE gives the *desired values. - Thomson (loc. cit. p. 319) estimated that for upper limits below 6.47/2 seven terms a t most would be needed to give results exact to four decimal places; by the method explained above it can be shewn that with this number of terms the error could not exceed z units a t the sixth decimal place. The fifteen-place table of Jas. Burgess [Trans. Roy. SOC. Edin. 39, 257 (1896)l was computed by a n altogether different method.

2858


(b) Approx. to J f ( x 2 ) . d x by series involving Je-m".j(x).dz (Used in computing F ( x ) Table XXVIII, see p. 2 8 7 4 ) . Replacing X by o in Eq. 105 and T by a x , multiplying through by z d / a l p . f ( z ) . d x , integrating between the limits o and and rearranging [ ~ x 2 ) . d x = ~ l K IdI (XZ ) d = x z ~ 8 . ~ ~ ~ - m 1 a z . f ( x ) . dRemainder x+a

(107)

The approximation consists in neglecting the "Remainder" of Eq. 107. If f ( x ) is of the form eBz (where B may be positive or negative, real imaginary or complex) a finite expression may be found in Table XIIa for the sum of the series arising from the lower limit of the integral on the right of (107); with other forms of f ( x ) the evaluation of this series may be so troublesome as to render the approximation undesirable. The error introduced by neglecting the "Remainder" of (107) is for practical purposes given by the value of its first item viz.

where M is the maximum absolute value attained by f ( x ) in the interval o < x < K 2 . Thus for given values of M and K , and any assumed value of a, an upper limit to the absolute error of the approximation can be found from Table XXVII; the largest convenient value of a consistent with the accuracy desired should be employed, as the number of terms of the series arising from the upper limit of the integral (in Eq. 107) that must be summed decreases If a be so chosen that the permissible error is with increase in K&. 2 X IO+.M~G or z X ~o-a.Md/al.rr, an upper limit to this number may be found from Table XV by writing K-\/a in place of 42 and entering on the line [odd]. A great advantage that this approximation enjoys over all the others is, that the number of terms to be summed increases very slowly with increase in K ; the reason is, that a slight lowering of the point E (Fig. 2 ) corresponds to a great increase in accuracy but only to a slight increase in the be considered neglinumber of terms to be summed. For example, if IO-~.M gible, four terms of the Evanescent Series will be sufficientwhen K = I , and five at most when K = I O ; if u=o.z be chosen in the first case and a = 0.0016 in the second. (c) Approx. to J e - b z "'.f(x2).dxby series involving ~ e - " ' " . f ( z ) . d r (Used in computing G ( x ) ,Table XXVII; see also p. 2869). Replacing X by z b d / a and T by ux in Eq. 105, multiplying through by zv'/ala.f(x).dx integrating between the limits o and K 2 , and rearranging

(109)

The approximation consists in neglecting the "Remainder" of (109) ; the series may be summed by the methods of approz.ii iii or iv (p. 2 8 2 5 ) depend-


2859

ing on the value of K/2b; all are available if f(s) is a member of Class A or B as defined in Table X, and all but appror.iv if it belongs to Class C. When f(x) = e'* integrals of the type here considered cannot be evaluated by means of approx.(b), as the f(x) of that approximation would be eCr-**'* and the integrals required are untabulated. I n this case, according as the series is to be evaluated by the method of approx. ti iii or zv, approx.(c) gives Eqs. I IO, I I I and I 12 respectively as expressions for the integral.

In Eq. I I O (to be used when 6 E/. < ~ / z < b a),d;is to be given the largest convenient value not exceeding 2 7 r E / ( n K + 4 b E ) where E is the ordinate assigned to the point E in Fig. 2. If E = o . 2 2 7 , from 4 to 6 terms of the Evanescent Series are needed to give the value of the integral with an error not exceeding 2 X 1o-~.JIK~4/ala. ~ ~ ~ = z - t " " ' . ~ ~ ~ ( ~ ~ / ~ ( b- )I)n+I.e-n2a2KZ . e C K ' C 4b2/(n2a2/4b2- C)

+

- ( d m ) . c o s e c 2b& (.\//8bC).eCK' (1 11) In Eq. 111 (to be used when 4 E / a < ~ / ~ b < 6 E / i rif) , E = 0 . 2 2 7 three or four terms of the Evanescent Series must be computed to evaluate the integral with an error not exceeding Iob.MK1.4\/X/2b. K

e C " " ~ " " . d x ~ ( ~ / z ~ ~ ) . e C " ' . F ( - \ / C K-2 (d/a/I6).tan ) d\/b2C (-\/;/ab) .ecK2.~ e - m 2 T Z K z ~ 4 ~ z / ( nC)2 ~ 2 / ~ b 2

+

(112)

I n Eq. 1 1 2 (to be used when ~ E / T < K / ~ ~ < ~ Eif/ TE =) 0, . 2 2 7 from two to four terms of the Evanescent Series must be computed to evaluate the integral with an error not exceeding 10-~.XK?.J/a/2b. Eqs. 1 1 0 - 1 1 2 were obtained from Eqs. jo, j z and 54 respectively; the value E = 0.2 2 ; is that of Fig. z as printed. Eq. I I O may be used for z E/. < K / z b < = without exceeding the error stated above, nine terms being required when K / z b = z E / n = 0.145; Eq. 111 may be used for z E / a < K / z E ) < 6 E I x with a maximum of nine terms; Eq. 1 1 2 on the other hand is valid only within the range stated after that equation. Thus no matter what the values of K and b may be, not more than nine terms of the Evanescent Series need be computed (six if the best formula be used) in order to ensure the postulated accuracy; values of E for other degrees of accuracy can be found from Table XIT. Eq. I I Z cannot be used when C is complex; all the others are valid whether C be positive or negative, real imaginary or complex. If C = o and b = a the integral is (I/z).G(K~),while if C = I the integral is H ( K , b l K ) , see Table S S I I , S o . I . If the approximation be based on [even]& instead of on [odd]Bc,the terms on the right of ( I 1 2 ) must be modified as follow:-Change the signs of the f i s t and third terms, replace tan hg cot in the second term and m by zn in the third, and add a fourth term viz. v'/;;.eCK'/ 4bC. The equation so modified is referred to on p. 2861 and p. 2877 as Eq. I ~ z a .

2860


The expression given after each equation for the upper limit to the error is deduced from Rule I, p. 2840; M stands for the maximum absolute value of e' when r lies between o and K 2 . When C is a positive constant M = e C K 2 , and if C K 2is large, Rule I much overestimates the possible error; in such a case Rule V (p. 2843) is more useful, it shews that for all three equations, no matter what the value of C K 2 may be or the form of f(r) (so long as it retains its sign in the interval o < z < K ~ the ) error cannot exceed one fifty-thousandth of the value of the integral if E = 0 . 2 2 7 , while if the value E = 0.18 be chosen (which involves the computation of at most two additional terms of the Evanescent Series) the error cannot exceed one millionth part of the value of the integral. For an example of this application of Rule V to a special case, see p. 2869.

(d) Approx. to f e"'.f(z).dr by series involving f cosh n z f ( x ) . d r (Used by H. G. Dawson' in 1898 to compute the integral of e'*.dr). From the expression for [dZ]€Ic in the second line of Table VIII, replacing X by z x d / T , multiplying through by ze"'.f(z).dx, integrating between the limits o and K , and rearranging:

The approximation consists in neglecting the series in the second line of ( I 13) ; in order t o facilitate the summation of the series in the first line, the smallest convenient value of T consistent with the accuracy desired should be chosen. T o find an approximate upper limit to the absolute error, the series neglected may be replaced by its first term, the cosine by unity, andf(z) by M vie. the greatest absolute value attained by f ( r ) while z increases from o to K ; this gives 2M.eK'-T.F ( K ) . An upper limit to the number of terms of the series in the first line of ( I 13) that must be computed, is the n found from 2 M . e K ' - T . F ( ~= ) (M/n~/n).e-n'"'/T,sinh(~nrK/.\//T) (1 14)

after giving T the smallest value that will make the expression on the left of (114) negligible in view of the accuracy desired. This approximation resembles approx.(a) in that the number of terms that must be computed increases very rapidly with increase in the value of K . Settingf(r) = I , the integral becomes that of ezz.dx, and for K = z the value T = 20 ensures accuracy in the sixth decimal place (Le. eight significant figai.es) when nine terms are computed; Dawson used T = 4 r 2 which (for K = 2 ) required 13 terms to attain the same accuracy. By the use of appror. (6) nine-figure values of this integral, no matter what the value of K may be, can be secured without computing more than seven terms of the Evanescent series of Eq. 134, see p. 2874. H. G. Dawson: Roc. London Math. SOC., 29, 519 (1898).

NCMERICAL EVALUATION OF INFINITE SERIES

2861

By settingf(x) = sin or cos bx, an approximation to the integral of ex'.sin or cos bx.dx. may be found; the number of terms required to hold the error below Iod.eK'.F(K)increases with increase in K from three when K is very small to five whenK =0.9 and seven when K = 1.1; for larger values of K , it is easier to evaluate these integrals by computing H(K,b/2) and K(K,b]2) as shewn in the Appmdix. (e) Use of the approximations in the construction of a Table In discussing the foregoing approximations it has been tacitly assumpd that the object of the computer is to find the numerical value of some single integral whose limits etc. have been specified; this is not quite the point of view when a Table of the integral in question is being constructed. In the latter case an endeavour would no doubt be made to proceed by means of Taylor's Series, or the methods of tabular integration, and the approximations would be employed only to furnish the relatively small number of accurate values nteded to prevent accumulation of errors; thus it might be well worth while to secure additional accurate values by recombining the terms of Evanescent Series already computed even although the limits etc. of these new integrals can not be arbitrarily selected but are fixed by the procedure used in evaluating them. The absolute values of the individual terms of the Evanescent Series of Eq. I Iza would be identical with those of the corresponding series of Eq. I I I if in the latter b2 were replaced by b2/4 while C and K 2 had the same values in both; thus the Evanescent terms which had already been used t o evaluate the integral of ecz'-62!r2.dx by (112a) could be used over again to evaluate the integral of ecz2~**'@*.dx by (111). For examples, see p. 2870 and p. 2877, KO such device is available with approx.(a) or approx. ( d ) ; and if it be adapted to approx.(b) the second integral will be of a different form from the first. If approx.(c) is being used in the construction of a Table, there is another way by which labour already expended can be utilized again. Like all the others of this Part, this approximation is based on the neglect of a Remainder; and the range over which each of its three subdivisions is valid, has been fixed by the assumption that the absolute value of the appropriate Remainder is unknown (but that it lies below some predetermined limit). The first item of the Remainder of (I IO), however, viz. 2~'a/~.Rl~:[odd]9c,is numerically equal to the integral itself for the same values of C and K but another value of b ; so that if the portion of the Table already constructed includes the value of the integral for this value of b, an approximation may be employed which neglects only Rlb and subsequent terms of the Remainder of (I IO) and requires the computation of fewer terms of the Evanescent Series than does the approximation of (I IO). This modification corresponds to choosing an auxiliary series (see approx. ii, p. 2826) on BG of Fig. 2 instead of on BE. Examples of the use of this method and of corresponding methods based on (I I I) and (I 12) are given on p. 2870 and p. 2877 ; no similar device is available in the case of approx.(a) and approx.(d); in the case of approx.(b) the information needed cannot be secured from the table under construction.

2862

W. LASH MILLER AND A . R . GORDOS

APPENDIX ON THE INTEGRAL l e z ' . d z AND RELATED INTEGRALS The group of integrals here to be considered includes the Probability Integral, the Fresnel Integrals, the Integral of ez*.dx and others illustrated in Table X X I and Eqs. 128 and 129; it will be shewn that all can be expressed in terms of two functions to which the symbols u and are given, and that all could be evaluated by means of two double-entry tables. As the tables in question have not been constructed, methods are described for evaluating the integrals in the absence of such tables. The practical advantages possessed by one particular pair of tables (viz. H and K ) in comparison with any other pair are then established, and are illustrated by the construction of Tables from which four-place values of the two Fresnel Integrals can be obtained for any value of the argument between o and m without recourse to graphical interpolation.

v

Sec. 14: The functions

If z

=

x

U and V

+ i y (where x and y are real and i = d q ) then ,

Since the integral on the left of (I 16)is a single-valued function of p and p, its real part vie. u(p,q) and its imaginary part viz. i v ( p , q ) must each of them be single-valued functions of p and q . This conclusion, on which all the subsequent argument depends, may be verified by differentiating' the expressions on the right of ( I I 7 ) and ( I IS),since the expressions for a*u/ap.aq and for aZv/ap.dp will be found to be independent of the order of differentiation. The only reason for introducing the complex quantity ( I 16) is that it and v(p,p) gives rise to functions of the real variables p and q, viz. u(p,q) which possess this property. Without assuming some relation between x and y, neither of the integrals on the right of ( I I 7 ) can be evaluated; any assumption may be made that permits x during the integration to assume all values between o and p , and y all values between o and p; it is obvious that the value of each integral taken by itself will depend on the assumption made, but since for given p and p u(p,q) has but one value, the difference between the values of the two integrals must be the same no matter what the assumption may be. If, for instance, it be assumed that y = o while x increases from o to p , and that then y increases from o to q while x retains the constant value p , 1

aU(p,q)/ap =

eP"y2.

COS

a p q ; aW(p,q)/ap.aq =

- 2 . e ~ ~ - - 9(q.cos ~

2pq

+ p.sin 2 p q ) .

XCMERICAL EVALUATION OF INFINITE SERIES

2863

U ( p , q ) = / 30 . d x - ep2.J:+.sin

(119)

apy.dy

1‘-

v ( p , q ) = ep2.

e “.cos z p y . d y

(120)

while if it be assumed that x = o while y increases from o to q, and that then x increases from o to p while y = q ,

U ( p , q ) = e+.Ji+.cos

(121)

+e

~ ( p , q =/:+.dy ) 0

0

zqx.dx

6

(122)

P

rK FIG.3

Paths of In:egra:ion:-Tlie numbers j.6 etc. are those of Tab. X U ; numbers in brackets are equation numbers. Ths letters are those of the text p. 2862-2868 except r K a n d s K which are those of Tab. XXI So. 9.

and the conclusion to be drawn from the preceding argument is that the expression on the right of (119)maybe equated to that on the right of ( r z ~ ) , a n d that the right of ( 1 2 0 ) may be equated to the right of ( 1 2 2 ) . If the successive values assumed by x and y during the integrations be represented by points on Fig. 3, the first assumption corresponds to the line Opd and the second to OqA; these lines are often spoken of as “paths of integration,” and the statement that U(p,q) is “independent, of the path” is only another way of saying that for any given p and q , U ( p , q ) has but one value. By integrating along different paths the differences (or sums) of Y large number of integrals may be expressed in terms of and I‘;for any line on Fig. 3 defines some relation between x and y, and the algebraic sum of the integrals defined by this relation gives the difference between the values of (or of as the case may be), at the points denoted by the extremities of the line. In general, if the relation be y = j(x),

u

u

v,

2864

W. L A S H MILLER A H D A. R. G O R D O N

Eqs. 1 2 3 and 1 2 4 contain between them four integrals; there are two forms of f(z) for which it is possible to eliminate' two of these integrals and combine the other two, these are:I. The$rst formf o r which elimination is possible i s x . f ( x ) = b2;for this form f'(s) = - b2/x2and the trigonometric factor of the integrand is either s i n zb2 or cos 2b2 i.e. is independent of z. The path of integration in this case is a rectangular hyperbola cutting the diagonal OB of Fig. 3 a t the point z = y = b Elimination gives

l~2z-b'12z + .dx = { U ( p , bZ/lp) -

U(a,€?/CY)}.COS 2b2 ( v ( p , be/p)

Jp$*e21-t4!21

.dx = ( U ( p , b 2 / p )

-

- I.'(cu,b 2 / a ) ) . s i n zb2

(125)

u(a,b 2 / a ) ) . s i n zb2

a

- { v ( p , b2/p)

- v ( a , b?/a)l.cos

2b2

(126)

+

T h e second f o r m f o r which eliminaiion i s possible i s f ( x ) = p x y; 2. this form corresponds to any straight line on Fig. 3 . Here j ' ( x ) = p, and elimination gives

=

U(P,b p + y )

- U(a,Pa+y)

+ P.V(P,P?J+Y)

- P.V(a,Pa+r)

(127)

and a similar equation for the integral containing s i n instead of cos in its integrand. I n these integrals the coefficients of x2 and x in the sine or cosine depend on those in the exponent; expressions in terms of u and V for analogous integrals where all four coefficients are independent of one another and of the limits of integration may be found by substituting wz+a for z and eliminating by means of the trigonometric formulas for the sines and cosines of sums. These expressions are (p.cos M s i n M)A eRz2+ 2S2 .cos ( 2 Tx2+zWx).dx = (cos N - /3..sin M )A Denominator (128) ( c o s A I - P . s i n M ) A v - (P.coshl+ s i n M ) AU ~e"f"~.~in(zrx2+2~z).dx= Denominator (129)

JP

u+

+

v

where to save printing the following abbreviations are used : I n general, of course, this is not possible; for instance no elimination is possible if y2 - z* = @,the path corresponding to which is a rectangular hyperbola asymptotic to OB of Fig. 3.

2865


AV= V ( PPP+Y) , - V(~,P~+Y) AlJ = U(p,&+y) - U(a,b + r ) ; P = ( p - a ) / ~ ; R =(I-/Y)W~;S= (I-~~)cYW-~~YW;T=~W ( Z L' Y; ~W + Y=) W ; M = za(aP+r);Denominator = w ( I + ~ ~ ) . ~ " - ~ ) ~ ' - ~ ~ ~ ~ - ~ * ~ By giving special values to a p y and w , the relatively simple forms of Table XXI may be obtained. On the other hand, if an integral of the most general form of Eqs. 128 and 129 be presented for evaluation, the numerical values of p , a , P, y , and w needed on the right of Eqs. 1 2 8 and 1 2 9 can be found from the numerical values of P , R, S, T and W in the given integral by means of the following equations': 2T.P = d R 2 f 4 T 2 - R zw2 = dR2+4T2 R p = UP a w.y = - aapw (82 1)w.a = (130)

+

pw + s

+

+

w

TABLEXXI: The functions

uand V

- U(-P,d = - U(-P, - d = U(P,-4 V(-p,a) = - V ( - p , - q ) = - V(P,--P)

1.

lJ(P,d

2.

V(p,n) =

3.

U(0,o) = V ( 0 , o ) = U(0,q) = V ( p , o ) =

4.

U(p,o) = p ' . d X ;U(p,Cc)= 0 ; V(0,q)=JE-%ix;V(p,oo)=& / 2

=

0

0

5.

l;'.cos

zBx.dx = eB2. U(K.B)

6. l a " . s i n 2 B x . d ~= eBz.V ( K , B-) eB'.J"''.dx 0

7.

l ~ - " . c o s 2Bx.d~=

9.

~ ~ ' . e zBx2.dx o r

e-".I/(B.K)

+ ~.V(TK,~K))/.\/~B?+I where z r 2 = 4@+1 + and = 4 4 B ' + 1 = (T.~(TK,SK)

I,

12.

2s2

I.

A : [odd or even] @c (f.u= e*'".sin or cos bzu+a) when gx= o can be reduced to the algebraic sum of the two integrals of Eqs. 128 and I 2 9 , each multiplied by a factor; in both of them P = &, R = C, S = W = o , T = b 2 / 2 . The Remainder integrals cannot be so reduced. See also Tab. XXII Nos. 5 and 6. Valid except when R = T =o, which gives

w

=o, so that

P

p and

y

are indeterminate.

w.

iXhli

LASH M ~ ~ . L EAUI) H .h.

w. wimm

'SABLEX X I : The functions c' and V (Continued) 1.3.

:I;

= 2''' +"',,siu or cos p~'+/ti;) can bc similnrly reduced; here P = l , 11=6 - i/&, zS=e+r/aKI, 21'=r, z W = h ; t,he Itemaindi~iutegrals cun hr handlrd sirnilrirly, using tlie suhstitutions of Tab. XXII No. 6 .

/o,ir/ or eironj.\e (P.h

f , a .4

f 'rrsicilrrbilitii oJ' c'(p,p) and V(p,q) Jor purposrs oJ" fubulalirin

If :,I rlw point

l ~ = p !,=ql , of Fig. 3 , ii line were mwted perpendicular to ttio (L,!,) pl:rnr and of length propurtionril to c'(plq), a n d similarly ;tt all other points of the figure, !hc surnrnil~of all (hese lines would lie in t.he U-mrfiicc~;tlir V-surfsee, thr sorfnee, ctc. might he constructed sinrilnrly. I l g . 4 shews:%portion of the ( U t V ) surface, vis. th:Lt for values of 7' and q within the dotted lines of Fig. 5 ; this surface was chosen for illustration rather !hail the u-surfircr o r the V-surfiLcr because u(p.qj v ( p , q ) dong the line of intersection of the siiril~ccwith thc plane x = ,q (see OB cos &.OLz, one of the Fresnel Integruls. As is a p of Fig. 4) is equd to 2

(u+vj

+

ptrrmt in Fig. 4 the

.t+ v)surface is churircterizrd by rapid changes of

(L'


2867

level; for small q’s, and p’s larger than those included in the figure, these changes soon became too great for graphical representation; but the differential equation for the contour lines viz. d q / d p = - sec. 4pq - t a n 4pq (from d.U(p.q) d.T/T(p,q) = 0) gives some idea of the form of the surface. The U-surface, the V-surface and the (V-V) surface have the same mountainous character; so that double-entry tables of any of these functions, which obviously describe the corresponding surface in detail, would have to contain a tremendous number of entries if interpolation were to be possible. The line of intersection of the U-surface (or the V-surface) with any plane perpendicular to the (z,y) plane, will in general’ cut through hills and valleys; the elevations along this line give the values of the integral whose path of integration is a straight line on the (x,y) plane, viz. the projection of the intersection; such integrals are therefore in general unsuitable for tabulation, and so are the integrals of Eqs. 128 and 129 (e.g. Nos. 6-13 of Table XXI) which are linear functions of the others. The Fresnel Integrals are the only members of this group for which tables have been constructed, and except the Brit. Ass. Table2 of 1926 which contains 2 0 0 entries for the first three cycles, none of them can be interpolated without the use of a graph (Cornu’s Spiral). On the other hand, if the projection of the intersection be an hyperbola for which zy = constant, the line of intersection (e.g. Fig. 4, 2pp = n/z and 2pq = 3 x / 2 ) will lie in the general direction of the ridges and valleys; and the corresponding integrals viz. H ( p , q ) and K(p,q) will be free from the property that makes tabulation of U ( p , q ) and V ( p , q ) impracticable.

+

Sec. 15 : The functions H ( p , q ) and K ( p , q ) The first of the two expressions for H(p,q) or K ( p , q ) given in Table X X I I corresponds to a double-entry table in which pq is constant for each column and p for each line; the values of both functions are positive throughout, both increase down the columns and H ( p , q ) decreases t o the right along the lines. Consideration of the second derivatives3 (see Table XXII) shews that H(p,q)can have no point of inflexion within a column, none in the lines for which po.92; while K ( p , q ) must have a maximum in every line for which p>0.92, and one point, of inflexion in every line for which p q = pd/I--p2, none in the others. The second expressions given in Table X X I I for the functions correspond to tables in which p q is constant for each column, and p / q for each line; thus in both

’ That is, except when p

= o or p = 0. Brit. Ass. Report, 1926, pg. 273. For instance, the second derivative of H with respect to q while p remains constant is proportional to z p H - E (Table X X I I ) , and the point to be settled is whether this quantitv will pass through zero while p remains constant and Q increases. If p = 0.92 and p = 0 , 2 = z p H (Tab. XXV S o . 9 ) ; as q increases both E and H decrease, hut the percentage decrease in H i s greater than in E as is obvious from the form of the integral in the first expression for H in Table X X I I . Therefore for all values of p > 0 , while p = 0.92, E > z p H , and in general there cannot be more than one value of q for each p a t which E = z p H . Since E - z p H has the same sign a t p = 0,and a t q = p when p < 0.92, there can be no point of inflexion for this value of p ; above p = 0.92 the signs are different, therefore there must be one point of inflexion and it has just been shewn that there cannot be more than one.

2868


LP

e"'.dx; K(p,o) =

3.

H(o,q) = K(o,q) =

0 ;H ( p , o ) =

4.

H(K,B2/K)

ez2- Ea 2%

5.

= cos 2B2.U(K,B 2 / K ) sin 2B2. V ( K , B 2 / K ) - ( d , / z ) . s i n zB2 A : [ o d d or even] @c (f.u = = times the right of No. 4 where K = b.\/t and 2B2 = bgx/&.

=

lK

.dx

4?r/2.

+

6. R,, may be substituted for A in No. 5 if gx be replaced by n x - g z ; Rnbmay be substituted for A in No. 5 if gz be replaced by n r f g x .

7 . Derivatives:Here E is written for ep*-Q1,H for H(p.q) and K for K(p,q). The quantity kept constant during the differentiation is printed as a subscript.

(aH/ap)q = E - 2 9 K ; (aH/aq), = - 2 p K ; (aH/ap)w= E (aK/ap), = 2 ~ H ; (aK/aq), = 2pH - E ; (aK/ap)pp= SEIp (azH/aq2), = - 2 p ( 2 p ~ E); (a2K/ap2)p = 2qE - 4p2K (a2Hlap2),,= 2 (p2+q2)E / p ; (a2K/ap2)pq = 2dp2+q2- 1)E/p2 pairs of tables the columns correspond to the hyperbolas of Fig. 5 , in the first pair the lines correspond to perpendiculars in the figure and in the second to rays from the origin. But although by tabulating H and K instead of u and the violent oscillations characteristic of the latter would be avoided, nevertheless the rapid increase in H(p,q) and K(p,p) with increase in p would introduce another difficulty for the interpolator; for instance in the column p q = 4x the value of H(p,q) would be 0.0691for p = q = 3.55 and 1.23 x 10' for p = 5.01. The best way to overcome this difficulty would be to tabulate the products of H(p,q)and K(p,q)into eq'-p2instead of the functions themselves, a t all events for p > q ; in such a table the entries 0.0691for ~ ' 3 . 5 5and 0.0800 for p = 5.01 would replace those given above. This device may be illustrated by the case of F ( x ) , Tab. XXVIII; of course to find the values of the integral of ez*.dz from the F ( z ) table, e'' must be found by interpolation of a table of exponentials', but by means of the multiplication theorem for exponentials interpolation in the latter table can be effected as closely as de-

v,

l J. w. L. Glaisher: [Camb. Phil. SOC.Trans., 13, 243 (1883))nine fi we8 from z = o z = 500; G. F. Becker and C. E. van Orstrand: [ HyperboEic Fundions, Bmithsonian Inst. (1909)l seven to nine figures from z = o t o z = 100; F. W. Newman, see p. 2874 f . note.

to

2869

NUMERICAL EVALUATION OF INFINITE GERIES

u

v,

sired. Similarly, by using tables of H and K to find' values of and the difficulty of interpolating the latter functions would be avoided; as there are addition theorems for sines and cosines and excellent tables2 of those Functions. Methods of evaluating H(p,y) and K(p,y) and for delimiting the values of p and q for which each method is available will now be explained; these are based on ap?rox.(c) p. 2 8 j 8 , and approx.(b) p. 2 8 j8;the use of approx. ( a ) and approx.(d) is in general not advisable, although for very small values of p the latter might prove useful (see p. 2861 and Table X X I Xos. 6 and 7 ) . By means of the relations given in Table XXII, any one of the four methods (viz. one based on appr0x.b and three on appr0r.c) can also be used to evaluate u ( p , y ) and v ( p , q ) and the integrals of Table S X I I and Eqs. 128 and 1 2 9 . The semi-convergent series obtained in the usual way by "integration-byparts" are too clumsy for practical use, except in the special cases that p = o (Probability Integral), p = y (Fresnel Integrals, see Table XXIII) or y = o (the Integral of esz.dx, see Table XXV).

Evaluation of H(p,q) and K(p,q) by means of approx. (c) and (e) To use a p p r o r . ( c ) , K must be replaced by p , C by I , and b2 by p'y? in Eqs. I 10-1 I 2 ; the left hand member of each of these equations then becomes H(p,q). If both sides of these equations be differentiated with respect to b as sole variable before the summations are made, the left hand member becomes twice K ( p , y ) ; as the differentiation leaves all the exponentials unchanged, the statements as to relative error and maximum number of terms made below hold for K(p,y) as well as for H ( p , y ) . Lower Boundaries for Eqs. I I I and I I Z are found by Rule V, p. 2843, assuming that the error must not exceed one three-millionth part of the quantity to be computed, and are entered in Fig. 5. C'pper Boundary:-The hyperbola y?-.p?= I j of Fig. 5 runs through values of p and p for which eP""=o.3 X IO+. Since for y z p , both f l ( p , y ) and K(p,g) are less than ePP-'?, this line constitutes a safe Boundary above which the values of both funct'ions will be negligible if 0.3 X IO-^ is the accuracy postulated in fixing the other Boundaries of the figure. Boundaries for (112):-1n this equation, H(p,y) is regarded as &/2py times R i : [ o d d ] S cwhere g x = o , t = p 2 and a = + / 4 p 2 y 2 ; Eq. 74 fixes ( R z : 8 c ) ; ( R 1 : e c )as an upper limit to the relative error. Introducing the values of the constants, and noting that e-'j=0.3 X IO+, this gives e-39?5e-1j, i.e. y = 2 . 2 4 as the lowest value of y with which Eq. I 1 2 can be relied upon to give the accuracy postulated. With this value of y only two terms of the Evanescent Series need be computed; with larger values of y the accuracy increases but so does the number of terms; when y = 4. j, five terms may be needed, and for q > 4 . 5 the approximations of (131) and (132) are preferable. From Table XXII Nos. I and 2 , C(P,q! = cos 2 p q . W p , q ) sin 2 p q . W p , q ) . v(p,q: = szn z p q . H ( p , q ) - cos z p q . K ( p , q ) &/2. The most complete are the fifteen-figure tables a t intervals of 'Youvelles tables trigonomktriques," 3 vols., (1916).

+

+

IO

sec. by H. Andoyer:

W. LASH MILLER AXD A. R . GORDON

2870

Boundaries for ( I I I) :-Here H(p,q) is regarded as d \ / K / z p q times A : [even]\kc where gx = T / Z , t = p z and a = ~ ~ / 1 6 p and ~ q ~Eq. ; 74 gives e-8gz= e--15, Le. q = 1.37, as the lowest value permissible. Three terms are needed when 9'1.37 and five when q = z . z 4 ; for q > 2 . 2 4 the approximation of Eq. 1 1 2 is preferable. Boundaries for (110):- Here H(p,q) is regarded as zdG times A:[odd]\kcwhere gx= z p y & and t = p L . To keep down the number of terms of the Evanescent Series that must be computed, a should be given the largest convenient value consistent with the ___ accuracy desired. Eq. 74 gives e --R?, 4p2a. P da = e-!; whence p & = ~ ( d @ + l 6 - 9 ) / 3 0 is the largest admissible value of p d a . With this value, five terms are needed if q = o , and seven if q = 1 . 3 7 ; if q > 1 . 3 7 the approximation of (111) is preferable.

1 4

4 2

0

4

2

6

PF I G .5

Extension of the Boundaries (see approx.e, p. 2861) :- The Remainder neglected in (112) is ( ~ \ / ? r / Z p g ) . ( - - ~ z : [ o d d ] \ k c R ~ : [ o d d ] \ kc . . .), and the quantity computed viz. f i ( p , q ) , is ( d ; / z p q ) . R 1 : [ o d d ] \ k c ,in all of which gx = 0, t = p 2 , a = n2/dp2q2and j ( u ) = e". But since, by Tab. VIII, R P:[odd]\kc would be equal to (~/a).Ri:[odd]\kcif in the latter q were replaced by 2q, the first item of the Remainder neglected in (112) is nothing but H ( p , d q ) . If a table were being constructed, and H(p$q) had already been computed, its value might be added to the approximate value of H(p,q) given by ( I I Z ) , thus reducing the error to that involved in the neglect of ( d r / d p q ) . ( R 3 . : [odcl]\kc - R a : [ o d d ] q c - . . .). For practical purposes this last expression may be replaced by its first term, which is nothing but H ( p J 3 q ) ,and by Rule l' this cannot exceed one three-millionth part of H(p,q) if q t 1.37. Thus if a table of H(p,q) were being constructed beginning with high values of q , the entries already computed might be used to correct the erroneous values given by Eq. I I Z when q < z . z 4 , so that this eqation might be ut,ilized for values of q as low as q = 1.3 7 . Thus the use of ( I I I ) might be dispensed with

+

+

+


2871

altogether, and the computation of five, four or three terms of the Evanescent of (111) might be replaced by that of two or one terms of the Evanescent of (112). The Boundary of (I I I ) can be extended in a similar manner from p = 1.37 to q=o.8c; in this case the saving may be considerable, since one to three terms of the Evanescent of (I I I) take the place of five to seven terms of the Evanescent of (110). The procedure consists in subtracting H(pJ3q) from the value of H(p,q)given by ( I I I). The corresponding correction to (I IO) replaces the Evanescent Series of which five to seven terms must be computed by one of which three to five terms suffice. This it does by assigning to 4; a value fixed by p & = ~(4q)/30 instead of the value stated to be permissible under “Boundaries for ( I I O ) , ” and then correcting the erroneous value so obtained for H(p,p) by adding H ( p , K- z p q d / a / Z p ) to it.

+

Evaluation of H(p,q) and K(p,q) by means of approx. (b) To use approx.(b), the K of Eq. 107 must be replaced by p; if then the f(z*) be replaced by e(’-’*)**.(cos z ~ z ?- ?.sin z q x 2 ) where y = q j p , the left hand member becomes u(p,y); while if the f(r?) be replaced by (4.~0sz?x2 sin zqz?), it becomes V(p,q). Boundaries‘ for (131) and (132):-If the value allotted to the constant a be such that, p d / n = o . q j , Rule I shews2 that the error of the approximation cannot exceed Io-e.(pq+p2).ep’-”, and Fig. 2 that not more thanfourtermsof the Evanescent Series need be computed no matter what the values of p and q may be. A small decrease in the value allotted to a , which involves the computation of an extra term or two, greatly increases the accuracy (see Table XIT’); for instance, if pd/a=o.30 the possible error is one hundredth of that when pd/a=o.45, and six terms will do. The expression for the Stationary Series, however, is so complicated that in practice the use of a p p r o r . ( b ) is to be recommended only for values of p and q such that both the cosh and sinh of qx/npd/a or K/Z& (whichever is the smaller) may be replaced by the exponential; the error introduced by this replacement will not exceed one three-millionth part of the cash or sinh if p d ; = 0.45 and both p and p are greater than 2 . 1 5 . The line 9’2.15 and the line p=z.15 thus constitute a Boundary below and t o the left of which the simplified form of a p p r o z . ( b ) is excluded if the postulated accuracy is to be attained; in practice, however, the Upper Boundary and the line p=4.5 may be taken, since it is at least as easy to compute five terms of the Evanescent of ( 1 1 2 ) as four terms of the Evanescent here considered. Assuming that p and q are large enough to permit the substitution of exponentials for the hyperbolic sines and cosines (see above) , the approximations to H(p,q) and K(p,y) which arise from npproz.(b) are as given in (131) and (132). These equations mere obtained as explained above from

+

For upper Boundary see p. 2869.

* Rule V is inapplicable, sincef(u) changes sign during the integration

2872

W. LASH MILLER AND A. R. GORDOS

(107) by using the last column of Table XIIa h’os. 5 and 6, and the relations between H and K given in Table X X I I ; the maximum error cannot exceed that given for U(p,q) and v(p,q)under “Boundaries for (1.11) and ( I , ~ z ) . ”

u,v,

Sec. 16: Three special cases (a) The Fresnel Integrals’ In a table of or S(z) proceeding by equal increments of z, interpolation is quite impossible after the first few entries, for as one goes down the table the maxima and minima succeed each other with increasing frequency; in Ignatowski’s Table, for example, the last two entries viz. c ( 8 . 4 ) = 0.4709 and c(8.j)= 0.5142 correspond to angles 1 5 2 degrees apart, and give no hint that c ( 8 . 4 2 6 ) = 0.4621. In tables proceeding by equal increments of the angle, i.e. of r x 2 / z , the maxima and minima are equidistant throughout; and if the tabular difference is small enough interpolation is possible, as in the B.A. six-figure Table which contains zoo entries for the first three cycles.

c(z)

TABLE XXIII: The Fresnel Integrals

s(z) =

l i i n $iry2.dy

=

3 - K-H -. 6

.

TX*

K+H

8x2

2

.\/R

2

san - - -.cos -

where K = K(x.\/?r/z, x&/2) and H = H ( x & / z , values of H and K, see Table XXIV.

2&/2);

for numerical

I n the tables of Gilbert and Ignatowski the value of C(z) isentered against the argument x‘ in Lommel’s table and the B. A. table, the value of C ( d z z / , ) is entered against z, 1.e. C ( l ) is entered against &/z. Gilbert: MBm. c o w . Acad. Brux. 3 1 (1863); Ignatowski: Ann. Physik, (4) 23,894 (1507); Lommel: Abh. Munch. Akad. ( 2 ) 15, 1 2 0 (1886); Brit. Ass. Report, p. 273 (1926).

Table XXIV gives (K+H)/d\/?r and(K- H)/& to four places as functions of z from z = o to z = 13, i.e. for the first forty-two cycles; after z = 2 , linear interpolation is sufficient to ensure the fourth decimal. For values of z greater than 13, ( K - H ) / ~ T = Oand (K+H)/.\/; = r/azexact to the fourth place. From this Table, by means of the relations given in Table YXIII, four-place values of c(z) and S(z) may be obtained for all values of z. This Table was computed from Gilbert’s and Ignatowski’s values of C(z) and S(z) by means of the relations given in Table X X I I I ; errors2 in the original tables The symbols C S H and K are defined in Tahle XXIII. Some of these, especially in Jahnke and Emde, are misprints. The true values are:c(0.1)= 0.1000;c(o.8) = 0.7225; c(1.1) = 0.7638; C(1.6) =0.6389; C(1.8) =0.333;; C(4.5) = 0.5260; c(5.I) = 0.4998; c(7.3) = 0.5190. 2


X

(K+W VG

0.00 .05 .10

TABLE XXIV; see Table XXIII X iK+H) IK-H) V'G 4; d+

iK - H)

,4078 ,3674 ,3306

2.00 2.10 2.20 2.30 2.40

0.1566 'I495 .I429 . I369 ,1315

,2972 .2671 ,2398 .2Ij+ ,1934

2.50 2.60 2.70 2.80 2.90

,1264

,0062

,1217

.oojs .ooso

,1057 ,1023 ,0992 ,0962 ,0933

0 .so00

0 .so00

,4520

.20

,4982 ,493I ,4855 .4760

.25 .30 .35 .40 .45

,4649 '4528 ,4399 ,4265 ,4129

.15

.1172

,1131 '1093

.50 .55 -60 .65 .70

,3992 . 3 8j 6 ,3723 ,3593 ,3446

,1736 ,1560 ,1402 ,1261 '1135

3.00 3.10 3.20 3.30 3.40

.75 .80 .85 .90 0.95

'3343 ,3225 ,3112 ,3003 ,2899

.I 0 2 2 ,0922 ,0833 ,0753 ,0682

3.50 3.60 3.70 3.80 3.90

0907 ,0883 .08j 9 ,0836 .08Ij

4.00 4.10 4.20 4.30 4.40

,0795 ,0776 ,0757 ,0740 ,0723

4.50 4.60 4.70 4.80 4.90

,0707 ,0691 ,0675 .0662 ,0449

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.60 1.70 1.80 1.90

,2800 ,2705

,2614 . 2528 ,2446 .2348 ,2295 ,2225

,2158 ,2094

,0618 ,0561 .OSIO

,0464 .0423 ,0386 ,0353 ,0323 ,0296 ,0272

,2034 . I922

.0250

. I820

,0182

,1727 0.1643

. 0 2 I3

,0154 0.0135

5.00 ,0636 5.10 ,0624 5.20 ,0412 5.30 ,0601 5.40 0.0589

For larger values of r , K --+ H 1/7rx;-K - H -

dF

2873

-

d+

- 0

are exact to four places.

0.0117

,0104 ,0091 .008 I

,007 I

,004 5 ,004 I

,0037 ,0033 .0030

7 .0025

,002

,0023 ,002 I

,0019 ,0018 ,0017

,0016 ,001s

,0014 ,0013 ,0012

,001I ,0010

,0009 ,0009 ,0009 ,0008 ,0008

IO008 ,0007 0.0007

U-R) VG

5.50 0 . 0 5 7 8 0.0006 .0006 5.60 ,0568 5.70 5.80 5.90

,0559

,0005

.os49 '0540

,0005

6.00 6.10 6.20 6.30 6.40

.os3 1

,0005

.os22 ,0514

,0004 .0004 ,0004 ,0004

6.50 6.60 6.70 6.80 6.90

,0490

,0461

,0004 ,0003

7.00 7.10 7.20 7.30 7.40

,0455 '0449 ' 0443 ,0437 .043 I

,0003 ,0003 ,0003 ,0003 ,0003

7.50 7.60

,0425 ,0420 ,0414 ,0408 0403

7.70 7.80 7.90

.Os06

'0498 ,0482

,0475 ,0468

I

8.00 8.10 8.20 8.30 8.40

,0398 '0393 ,0388 ,0384 ,0379

8.50 9.00 9.50 10.00 10.50

,0375 ,0354 ,033s ,0318 ' 0303

.ooos

,0004 ,0004 ,0004

.0002 .0002 ,0002

,0002 ,0002 .0002

,0002 ,0002 .0002

.0 0 0 2 ,0002 ,000 I ,000 I ,0001

.000 I

,0001 11.00 ,0289 ,000 I 11.50 , 0 2 7 7 , 0 0 01 12.00 ,026 j ,000 I 12.50 . o a j j 13.00 0.0245 o.oooo

2874


were found by differencing H and K (to find them by differencing the original tables was of course quite impossible) and were corrected by computation as explained below; from x = o to z = 1.5 Gilbert's tabular interval was halved, partly by interpolation and partly by independent computation of the series. For the first three cycles the B.A. Table gives six-place values; ten-figure values might be obtained, if needed, from the twelve-place Bessel Functions (B.A. Report for 1925 p. 2 2 1 ) as explained on p. 273 of the Report for 1926. For higher values of 2, Eqs. 131 and 132 may be used; for this purpose both p and p must be replaced by x & / z , whereupon the left hand members become the H and K respectively of Tables XXIII and XXIP. Ezample24 -Thelast entryin theB.A. T a b l e g i v e s C ( 4 G ) =0.580389andS(d\/40/n) =0.461646. Choosing a = o . o ~ ,seven terms of (131) give H = 0.0769 5320 488 and seven terms of (132) give K = 0.0808 7033 962 with a possible error well under IO-". From these, = 0.46164 57788, which agree to the last place C(d,/,) =o.58038 89720 and S(d\/qo/~) with the figures obtained by adding (in each case) twenty-two 12-figure Bessel Functions.

(16b)The function F(x) = e-'z./:y2.dy 0

The advantage which a table of F ( x ) has over one of the integral' of ez*.dx consists in the facility with which the former may be interpolated, as has already been pointed out. The function F(z) occurs in the P-approximations to @ E when gx=o and f(u) =ewbzu (see Table X I Nos. 6 and 7, and Ex. 7) and in the Original Integral when f(u) = ~ / - \ / b q , F(db"u) or (see Table XI11 Kos. 2' 5 and 6, and Ex. 1 1 , 1 2 and 17); also in the Poisson Form of 0~(see Eqs. 3 j and 36) and in the Poisson Form of A,u when F(X) is any odd integral power of X. The integral of F(x).ds occurs in the Poisson Form of h . when ~ F(X) is any even integral power of A ; it cannot be reduced to known forms and, is untabulated. The integral of Eq. 133, viz. twice A : [even]@s( I), needed in finding the error of the P-approximation to @.P (see p. 2824) may be evaluated by replacing F(z) by the semiconvergent series of Table XXV S o . 3, since only small values of T need be considered; this gives:

The numerical values of F ( x ) given in Table XXT'III were obtained as follows:- From z = o to x = 2 from Dawson's six-figure Table? of the integral of er*.dx and Newman's Table3 of e -'. From z= z to x = 4 nine-figure values at intervals of 0.2, and from x = 4 to x = 6 ten-figure values a t intervals of 0.5, were computed by Eq. 134.

F(x)-+( d K / z ) .e -'*.tun

K/Z

4; - 2 d/ala.Be -m'ar*/(mZu- I)

(134)

For tables to convert the first function into the other, see footnote p. 2868. * H. G. Dawson: R o c . London Math. SOC.,29, j19 (1898). F. W. Newman: Camb. Phil. Soc. Trans., 13, 145 ( 1 8 8 3 ) ;12-18 decimalp1aces;intervals of .ooI from x = 0 t o 1j.35, of ,002 to 17.30, of ,005 to 27.64, of .OI to 37.0,

NUMERICAL EVALUATIOS OF INFINITE SERIES

2875

TABLE XXV: The function F

3.

=

4- 1/4x3

1/22

+ 3/8x5 f

15/16x'

+ 105/32x9 f . . . (semi-

convergent) 4.

F(iz) =

i,ezz.~~-u9.dy

5.

d.F(z)/dz =

I

- zz.F(z)

0

d;- F ( d : )

6.

/F(dG).dx

9.

.It z = 0.9241 3887, F ( z ) has its maximum value viz. 0.5410 4 4 2 2 .

IO.

=

1/22

=

'kt z = 1.5019;6, F ( z ) hasa point of inflexion; here F ( z ) = z/(zz? -

I) =

0.427686.

which may be obtained from Eq. IO; by writing y for z and z for K and then replacing f(y2) by e -(*' -") . In each computation n was given the value o.I/z*, so that the error of the approximation was always less than j X 10-l~; seven terms of the Evanescent Series were needed to ensure accuracy in the tenth decimal place. From x = 6 to z= I O at intervals of 0 .j, and for z= I I and z= 1 2 , ten-figure values were computed by the semi-convergent series of Table XXV S o . 3 ; at z= 6 the value given by ten terms of the series agreed to the tenth decimal place with that obtained from (134), at z= 1 2 only six terms were needed. Intermediate values were obtained by means of Eq. 13j, which is nothing but Taylor's theorem, F ( z + o . o ~ )- F ( z ) = o . o ~ [ dF ( z ) / d z ] o . o o o j [ c ~ F ( z ) / d z 2 ] . . . (135) = (10-9/3).(29998000 - 2999752 f 2 0 0 0 X 2 - IOZ3) - (10-'//3).F(2).(29998j f 59994000Z - 599940X2 4 0 0 0 2 ~- 2 0 2 ~ ) and similar equations for F ( z f o . o j ) etc. when z > 4 . The two power-series were computed from their differences, on the adding machine; Eq. 13 j proved a rapid means of interpolation as well as an independent check on the accuracy of the nine and ten-figure numbers.

+

+

+

This form occurs as P : [ o d d ] * c withf(u) = ! 6 C / n 2times the H of Eq. 89, in computing the concentration of hydrogen dissolved in the catholvte during the electrolysis of acid copper sulphate solutions by currents above the lirnitingcurrent. The relation given above mav be established bv expanding F( dTf)by S o . z , integrating term by term and replacing thedefinite integrals so obtained by their known values from de Haan: loc. cit., Table VII).

2876

W. LASH MILLER AND A. R. OORDON

The integral of F(x).dx might be obtained by tabular integration of Table XXVIII, computing as a check a few points by the integral of Table XXV Xo. 2 or No. 3, or by Eq. 136. For large values of the upper limit K , the integral approaches infinity; for these high values it would be better to tabulate JF(x).dz - (1/2).log K which becomes constant.

m(m2Ia- I ) Eq. 136 was obtained by integrating (134); its error (see Eq. 108) is less than 2KdL

2K&

(1/8a).JG(y2).dy 0

which in turn, if Z K . \ / ~5

I,

is less than (1/8a)./G(y).

dy; Table XXVII shews that if ~ . \ / / a 5 0.j, this latter quantity is less than z X ro-’.P. If a be given the value 0.2 j / P , four terms of the Evanescent Series (via. the first on the right of Eq. 136) are sufficient to ensure accuracy in the sixth decimal place; the sum of the other series may be found from Table XVIII No. 7. Ezarnple 25:-To

evaluate

L1

F(x).dz.Choosing a = 0 . 1 6 the error of the approximation

will be less than 10-8, The value of the first term on the right of (136) is 0.66185 5656; the sum of six terms of the Evanescent Series is - 0 . 6 1 2 2 2 7917. By Table XVIII No. 7, the sum of the other series is given by C/z-((rr/4) . tan 5 r r / 4 + $ ( ~ . 5 ) - o . j $ ( r . z 5 ) = 0 . 3 2 ~ 9 3076; the value of $ ( 2 . j ) viz. 0.70315 6 6 4 1 , was taken from the B.A. table of the logarithmic derivate of the Gamma-Function; that of $ ( I . z j ) viz. -0.22745 3532, was obtained from Legendre’s table of loglor (tabular difference 0.005) by means of the formula 0.005$(1.25) =2.30258 5 0 9 3 ( A - A 2 / 2 + A 3 / 3 -A’/4) The value so found for the integral is 0.3697 2082. which is the same as that obtained by integrating Table X X V No. z between the limits o and I . By choosing the value 1/9 for a, interpolation in the logr table might have been avoided, but two more terms of the Evanescent would have had to be computed.

(16c) The functions G(x) and /G(x).dx (defined in Tab. XXVI, No. 1) Table XXVI Nos. 2 and 3 follow from Eqs. 44, 45 and 47 if in the latter f(u)= I . KO.4 was obtained by integrating by parts the integral on the right of KO.I ; it is useful in computing G(z) when z is small. 9 0 s . 5 and 6

TABLE XXVI: The Function G I.

G(x) =

-.dy = z.\//z.e-“a’L - 4s

j’E2

2.

Rna:[odd or ettenJQc(1)

3.

Rnb:[odd

4.

G(x)

5.

.\/r.G(x)

6.

.\//a.G(qx)

=

*b>2y

= (n,-sr/8u.\/x).G(4a2at/na--92*) or even]Oc(~)= (n~+gz/8a.\/?r).G(4a’at/~ia+yrc’)

( x ” / ’ / a * ) ~ e ~ ~ ‘ ’ ” . ( 1 - 3 r c / ~ ? r ~ + 3 . ~ . r ~ / 4 ~ ~. ~. .) 3.j.~~~/8~~+

+ n2/6 - .\/% -&-n2”/n’ - .\/,.zn.G(x/n?) = x - n2/6 + z.z(-~ ) ~ ~ + ‘ . e - ~-’ ~&.zm.G(4z/rn2) /n?

=

x/z

71=

2

n=z

NUMERICAL EVALCATION OF INFINITE SERIES

2877

are special cases of ( I I 2u) and ( I I I ) respectively; to obtain No. 5 from (I IZU) replace K~ by x and b2 by rZ7 and then take the limit when C+O, noting that when y+o F(y)+y; to obtain Xo. 6 from (111) replace K 2 by x and b2 by r2/4, and then let C+o. From Table XXVI KO. 4 it follows that for x 5 0.4, G(z)

numerical evaluation of isfinite series and istegrals ... - ACS Publications

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