Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.5 BESSEL FUNCTIONS

Prove the expression (18.91).

If we multiply (18.90) byxJν(αmx) and integrate fromx=0tox=bthen we obtain
∫b

0

xJν(αmx)f(x)dx=

∑∞

n=0

cn

∫b

0

xJν(αmx)Jν(αnx)dx

=cm

∫b

0

Jν^2 (αmx)xdx

=^12 cmb^2 J′^2 ν(αmb)=^12 cmb^2 J^2 ν+1(αmb),

where in the last two lines we have used (18.88) withαm=α=β=αn, (18.89), the fact
thatJν(αmb) = 0 and (18.95), which is proved below.

Recurrence relations

The recurrence relations enjoyed by Bessel functions of the first kind,Jν(x), can

be derived directly from the power series definition (18.79).

Prove the recurrence relation

d

dx

[xνJν(x)] =xνJν− 1 (x). (18.92)

From the power series definition (18.79) ofJν(x)weobtain

d

dx

[xνJν(x)] =

d

dx

∑∞

n=0

(−1)nx^2 ν+2n

2 ν+2nn!Γ(ν+n+1)

=

∑∞

n=0

(−1)nx^2 ν+2n−^1

2 ν+2n−^1 n!Γ(ν+n)

=xν

∑∞

n=0

(−1)nx(ν−1)+2n

2 (ν−1)+2nn!Γ((ν−1) +n+1)

=xνJν− 1 (x).

It may similarly be shown that

d

dx

[x−νJν(x)] =−x−νJν+1(x). (18.93)

From (18.92) and (18.93) the remaining recurrence relations may be derived.

Expanding out the derivative on the LHS of (18.92) and dividing through by

xν−^1 , we obtain the relation

xJν′(x)+νJν(x)=xJν− 1 (x). (18.94)

Similarly, by expanding out the derivative on the LHS of (18.93), and multiplying

through byxν+1, we find

xJν′(x)−νJν(x)=−xJν+1(x). (18.95)

Adding (18.94) and (18.95) and dividing through byxgives

Jν− 1 (x)−Jν+1(x)=2Jν′(x). (18.96)