Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

Using de Moivre’s theorem, expiθ=cosθ+isinθ, we then obtain

exp(ixsinθ)=cos(xsinθ)+isin(xsinθ)=

∑∞

m=−∞

Jm(x)(cosmθ+isinmθ).

Equating the real and imaginary parts of this expression gives

cos(xsinθ)=

∑∞

m=−∞

Jm(x)cosmθ,

sin(xsinθ)=

∑∞

m=−∞

Jm(x)sinmθ.

Substituting these expressions into (18.100) then yields

I=

1

π

∑∞

m=−∞

∫π

0

[Jm(x)cosmθcosnθ+Jm(x)sinmθsinnθ]dθ.

However, using the orthogonality of the trigonometric functions [ see equations (12.1)–
(12.3) ], we obtain

I=

1

π

π

2

[Jn(x)+Jn(x)] =Jn(x),

which proves the integral representation (18.99).

Finally, we mention the special case of the integral representation (18.99) for

n=0:

J 0 (x)=

1

π

∫π

0

cos(xsinθ)dθ=

1

2 π

∫ 2 π

0

cos(xsinθ)dθ,

since cos(xsinθ) repeats itself in the rangeθ=πtoθ=2π. However, sin(xsinθ)

changes sign in this range and so

1

2 π

∫ 2 π

0

sin(xsinθ)dθ=0.

Using de Moivre’s theorem, we can therefore write

J 0 (x)=

1

2 π

∫ 2 π

0

exp(ixsinθ)dθ=

1

2 π

∫ 2 π

0

exp(ixcosθ)dθ.

There are in fact many other integral representations of Bessel functions; they

can be derived from those given.

18.6 Spherical Bessel functions

When obtaining solutions of Helmholtz’ equation (∇^2 +k^2 )u= 0 in spherical

polar coordinates (see section 21.3.2), one finds that, for solutions that are finite

on the polar axis, the radial partR(r) of the solution must satisfy the equation

r^2 R′′+2rR′+[k^2 r^2 −(+1)]R=0, (18.101)