MATHEMATICAL PROBLEM PAGE Directed by EDWARD L. HAENISCH Montana State College, Bozeman, Montana 2.
PROBLEMS
k t a = 8 = x.
From (2) we have:
O S ~ sina% - (1 - COS%)
COS2z = C
1. Hermitian polynomials are often defined by the equation : n(n - 1 ) ( 2 ~ ) " - ~ Hn(x) = (2.x)" 1
= cosZx
'/dcos 2x
+ 1) = COS?~
since, sin%
Show that this definition is equivalent to the one given in equatiou (119), p. 383. Obtain the value of
+ +easar = 1.
for sin 0 = sin 2 s = 0. I n a similar fashion s i n 4 = 1/$(1 - cos 2%)and the inte-
Rely)
--a,--,.
2. Use the above definition of the Hermitian polynomial to prove equations (120) and (121), p. 383. 3. Show that +.(x) = e"'IZ. H&), for n = 1,isan actual solution of the S. equation used at the start of the article:
gral also evaluates to
and
r. 2
sin mx cos nrdx =
f *
sin (m
+ n)&x +
This demonstrates that the form determined from the arbitrary, = eP'/2 # ( x ) , actually satisfies the original A cos (m - 4% Z(m - n) equation. =-1+1-1+1=0. 4. Trace the curve corresponding to &, i. e., y = The other integrals can he handle? hy a like pmedure. 6 = e4'/' (he- 2). (Hint: Refer to the Problem .. Page for May in which "curve tracing" is discussed.) mw*. mrx = 2A; sin - ; + A = -2Ai sin By what factor does the curve you have obtained differ 4. d d from the one for n = 2 in Figure 20, p. 384? Wby was mwr = 4A' sin' this factor introduced? d 5. Evaluate the integral, A' e4'&, by the expansion of the function into a seqies and the subsequent integration of each term. 6. Find the center of gravity of a Bemisphere whose density varies as the distance from the base.
+
1.
SOLUTrONS TO PRORLEMS IN JULY x (a) Let ar = B = - From (I),* sin 2' X
X
+ s i n - cos2 2
4As 2
5.
X X M S ~ ~ X = ~ S ~ - C O S -
2
(1
+ sinx 2lC
- cosa 2E)
= iuAeh.* - iaJlc-imz
_
- a2Bc-im~
= -a%Ash~
dz'
= 1, we have cos x = cos*
or. 2 cosx
*
!?
X
case -X2
~f
dr
2
(h) Let ar = 8 = - From (2),cm 2' Since
4A' If - d = 1, then 2A = 2 + = Aeim + ~ ~ - i o =
= - d.
ISSUE
4% d
= -a2(Aeiez + BL-im%)
Substituting in the difierential equation: Bk-iw) 8r'p E(A& + Be-i= ) = 0 -af(Aei.x
X
-2
+
+
: = 1 + COS x 2
( c ) I f x = a + B a n d y = a - B,wehavea='/r(x+y)and = '/& - y). From (1) sin x = sin l/*(x y) cos '/l(x - y) cos '/dz y) sin I/&-y)
a
sin y = sin 1/.(x sin x
+
+
+ Y)cos ' / d r - y) - cos '/dx
+ sin y = 2-sin '/& + y) cos
+
+ Y)sin V ~ X - Y )
dN N
6. d~ = - y ~ d t ; - = --,at;
- y)
In N = -yt
+c
N when t = 0. N = NOor C = In NO. Thus In - = -yt Ne
* This number refers to a formula given on the July page. 392
393 N = Nae-TI; d N = -yNoe-%t. Using the formula for average life.
A table of definite integrals gives, F o r n = l a n d z = t , r = - 7 . - 1= - 1
r2
Y'
