Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.5 BESSEL FUNCTIONS

in subsection 18.1.2. The generating function for Bessel functions of integer order

is given by

G(x, h)=exp

[

x

2

(

h−

1

h

)]

=

∑∞

n=−∞

Jn(x)hn. (18.98)

By expanding the exponential as a power series, it is straightfoward to verify that

the functionsJn(x) defined by (18.98) are indeed Bessel functions of the first kind,

as given by (18.79).

The generating function (18.98) is useful for finding, for Bessel functions of

integer order, properties that can often be extended to the non-integer case. In

particular, the Bessel function recurrence relations may be derived.

Use the generating function to prove, for integerν, the recurrence relation (18.97), i.e.

Jν− 1 (x)+Jν+1(x)=

2 ν

x

Jν(x).

DifferentiatingG(x, h) with respect tohwe obtain

∂G(x, h)

∂h

=

x

2

(

1+

1

h^2

)

G(x, h)=

∑∞

n=−∞

nJn(x)hn−^1 ,

which can be written using (18.98) again as

x

2

(

1+

1

h^2

)∑∞

n=−∞

Jn(x)hn=

∑∞

n=−∞

nJn(x)hn−^1.

Equating coefficients ofhnwe obtain
x
2

[Jn(x)+Jn+2(x)] = (n+1)Jn+1(x),

which, on replacingnbyν−1, gives the required recurrence relation.

Integral representations

The generating function (18.98) is also useful for derivingintegral representations

of Bessel functions of integer order.

Show that for integernthe Bessel functionJn(x)is given by

Jn(x)=

1

π

∫π

0

cos(nθ−xsinθ)dθ. (18.99)

By expanding out the cosine term in the integrand in (18.99) we obtain the integral

I=

1

π

∫π

0

[cos(xsinθ)cosnθ+ sin(xsinθ)sinnθ]dθ. (18.100)

Now, we may express cos(xsinθ) and sin(xsinθ) in terms of Bessel functions by setting
h=expiθin (18.98) to give

exp

[x

2

(expiθ−exp(−iθ))

]

=exp(ixsinθ)=

∑∞

m=−∞

Jm(x)expimθ.