Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.13 EXERCISES

[ You will find it convenient to use

∫∞

−∞

x^2 ne−x

2

dx=

(2n)!

√

π
22 nn!
for integern≥0. ]
18.9 By initially writingy(x)asx^1 /^2 f(x) and then making subsequent changes of
variable, reduce Stokes’ equation,

d^2 y

dx^2

+λxy=0,

to Bessel’s equation. Hence show that a solution that is finite atx=0isa

multiple ofx^1 /^2 J 1 / 3 (^23

√

λx^3 ).
18.10 By choosing a suitable form forhin their generating function,

G(z, h)=exp

[

z

2

(

h−

1

h

)]

=

∑∞

n=−∞

Jn(z)hn,

show that integral repesentations of the Bessel functions of the first kind are

given, for integralm,by

J 2 m(z)=

(−1)m

π

∫ 2 π

0

cos(zcosθ)cos2mθ dθ m≥ 1 ,

J 2 m+1(z)=

(−1)m+1

π

∫ 2 π

0

cos(zcosθ) sin(2m+1)θdθ m≥ 0.

18.11 Identify the series for the following hypergeometric functions, writing them in
terms of better known functions:

(a) F(a, b, b;z),

(b)F(1, 1 ,2;−x),

(c) F(^12 , 1 ,^32 ;−x^2 ),

(d)F(^12 ,^12 ,^32 ;x^2 ),

(e) F(−a, a,^12 ;sin^2 x); this is a much more difficult exercise.

18.12 By making the substitutionz=(1−x)/2 and suitable choices fora,bandc,
convert the hypergeometric equation,

z(1−z)

d^2 u

dz^2

+[c−(a+b+1)z]

du

dz

−abu=0,

into the Legendre equation,

(1−x^2 )

d^2 y

dx^2

− 2 x

dy

dx

+(+1)y=0.

Hence, using the hypergeometric series, generate the Legendre polynomialsP(x)
for the integer values=0, 1 , 2 ,3. Comment on their normalisations.
18.13 Find a change of variable that will allow the integral

I=

∫∞

1

√

u− 1

(u+1)^2

du

to be expressed in terms of the beta function, and so evaluate it.
18.14 Prove that, ifmandnare both greater than−1, then

I=

∫∞

0

um

(au^2 +b)(m+n+2)/^2

du=

Γ[^12 (m+1)]Γ[^12 (n+1)]

2 a(m+1)/^2 b(n+1)/^2 Γ[^12 (m+n+2)]

.