Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

Finally, subtracting (18.95) from (18.94) and dividing byxgives

Jν− 1 (x)+Jν+1(x)=

2 ν

x

Jν(x). (18.97)

Given thatJ 1 / 2 (x)=(2/πx)^1 /^2 sinxand thatJ− 1 / 2 (x)=(2/πx)^1 /^2 cosx,expressJ 3 / 2 (x)

andJ− 3 / 2 (x)in terms of trigonometric functions.

From (18.95) we have

J 3 / 2 (x)=

1

2 x

J 1 / 2 (x)−J′ 1 / 2 (x)

=

1

2 x

(

2

πx

) 1 / 2

sinx−

(

2

πx

) 1 / 2

cosx+

1

2 x

(

2

πx

) 1 / 2

sinx

=

(

2

πx

) 1 / 2 (

1

x

sinx−cosx

)

.

Similarly, from (18.94) we have

J− 3 / 2 (x)=−

1

2 x

J− 1 / 2 (x)+J−′ 1 / 2 (x)

=−

1

2 x

(

2

πx

) 1 / 2

cosx−

(

2

πx

) 1 / 2

sinx−

1

2 x

(

2

πx

) 1 / 2

cosx

=

(

2

πx

) 1 / 2 (

−

1

x

cosx−sinx

)

.

We see that, by repeated use of these recurrence relations, all Bessel functionsJν(x)ofhalf-
integer order may be expressed in terms of trigonometric functions. From their definition
(18.81), Bessel functions of the second kind,Yν(x), of half-integer order can be similarly
expressed.

Finally, we note that the relations (18.92) and (18.93) may be rewritten in

integral form as

∫

xνJν− 1 (x)dx=xνJν(x),

∫

x−νJν+1(x)dx=−x−νJν(x).

Ifνis an integer, the recurrence relations of this section may be proved using

the generating function for Bessel functions discussed below. It may be shown

that Bessel functions of the second kind,Yν(x), also satisfy the recurrence relations

derived above.

Generating function

The Bessel functionsJν(x), whereν=nis an integer, can be described by a

generating function in a way similar to that discussed for Legendre polynomials