Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

18.24 The solutionsy(x, a) of the equation

d^2 y

dx^2

−(^14 x^2 +a)y=0 (∗)

are known as parabolic cylinder functions.

(a) Ify(x, a)isasolutionof(∗), determine which of the following are also

solutions: (i)y(a,−x), (ii)y(−a, x), (iii)y(a, ix)and(iv)y(−a, ix).

(b) Show that one solution of (∗), even inx,is

y 1 (x, a)=e−x

(^2) / 4
M(^12 a+^14 ,^12 ,^12 x^2 ),
whereM(α, c, z) is the confluent hypergeometric function satisfying
z
d^2 M
dz^2
+(c−z)
dM
dz
−αM=0.
You may assume (or prove) that a second solution, odd inx,isgivenby
y 2 (x, a)=xe−x
(^2) / 4
M(^12 a+^34 ,^32 ,^12 x^2 ).
(c) Find, as an infinite series, an explicit expression forex
(^2) / 4
y 1 (x, a).
(d) Using the results from part (a), show thaty 1 (x, a) can also be written as
y 1 (x, a)=ex
(^2) / 4
M(−^12 a+^14 ,^12 ,−^12 x^2 ).
(e) By making a suitable choice foradeduce that

1+

∑∞

n=1

bnx^2 n

(2n)!

=ex

(^2) / 2

(

1+

∑∞

n=1

(−1)nbnx^2 n

(2n)!

)

,

wherebn=

∏n

r=1(2r−

3

2 ).

18.14 Hints and answers

18.1 Note that taking the square of the modulus eliminates all mention ofφ.
18.3 Integrate both sides of the generating function definition fromx=0tox=1,
and then expand the resulting term, (1 +h^2 )^1 /^2 , using a binomial expansion. Show
that^1 /^2 Cmcan be written as [ (−1)m−^1 (2m−2)! ]/[2^2 m−^1 m!(m−1)! ].
18.5 Prove the stated equation using the explicit closed form of the generating function.
Then substitute the series and require the coefficient of each power ofhto vanish.
(b) Differentiate result (a) and then use(a) again to replace the derivatives.
18.7 (a) Write the result of using Leibnitz’ theorem on the product ofxn+mande−xas a
finite sum, evaluate the separated derivatives, and then re-index the summation.
(b) For the first recurrence relation, differentiate the generating function with
respect tohand then use the generating function again to replace the exponential.
Equating coefficients ofhnthen yields the result. For the second, differentiate the
corresponding relationship for the ordinary Laguerre polynomialsmtimes.
18.9 x^2 f′′+xf′+(λx^3 −^14 )f= 0. Then, in turn, setx^3 /^2 =u,and^23 λ^1 /^2 u=v;thenv
satisfies Bessel’s equation withν=^13.

18.11 (a) (1−z)−a.(b)x−^1 ln(1 +x). (c) Compare the calculated coefficients with those
of tan−^1 x.F(^12 , 1 ,^32 ;−x^2 )=x−^1 tan−^1 x.(d)x−^1 sin−^1 x. (e) Note that a term
containingx^2 ncan only arise from the firstn+ 1 terms of an expansion in powers
of sin^2 x; make a few trials.F(−a, a,^12 ;sin^2 x)=cos2ax.
18.13 Looking forf(x)=usuch thatu+ 1 is an inverse power ofxwithf(0) =∞and
f(1) = 1 leads tof(x)=2x−^1 −1.I=B(^12 ,^32 )/

√

2=π/(2

√

2).