Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.5 BESSEL FUNCTIONS

generality. The equation arises from physical situations similar to those involving

Legendre’s equation but when cylindrical, rather than spherical, polar coordinates

are employed. The variablexin Bessel’s equation is usually a multiple of a radial

distance and therefore ranges from 0 to∞.

We shall seek solutions to Bessel’s equation in the form of infinite series. Writing

(18.73) in the standard form used in chapter 16, we have

y′′+

1

x

y′+

(

1 −

ν^2

x^2

)

y=0. (18.74)

By inspection,x= 0 is a regular singular point; hence we try a solution of the

formy=xσ

∑∞

n=0anx

n. Substituting this into (18.74) and multiplying the resulting

equation byx^2 −σ,weobtain

∑∞

n=0

[

(σ+n)(σ+n−1) + (σ+n)−ν^2

]

anxn+

∑∞

n=0

anxn+2=0,

which simplifies to

∑∞

n=0

[

(σ+n)^2 −ν^2

]

anxn+

∑∞

n=0

anxn+2=0.

Considering the coefficient ofx^0 , we obtain the indicial equation

σ^2 −ν^2 =0,

and soσ=±ν. For coefficients of higher powers ofxwe find

[

(σ+1)^2 −ν^2

]

a 1 =0, (18.75)

[

(σ+n)^2 −ν^2

]

an+an− 2 =0 forn≥ 2. (18.76)

Substitutingσ=±νinto (18.75) and (18.76), we obtain the recurrence relations

(1± 2 ν)a 1 =0, (18.77)

n(n± 2 ν)an+an− 2 =0 forn≥ 2. (18.78)

We consider now the form of the general solution to Bessel’s equation (18.73) for

two cases: the case for whichνis not an integer and that for which it is (including

zero).

18.5.1 Bessel functions for non-integerν

Ifνis a non-integer then, in general, the two roots of the indicial equation,

σ 1 =νandσ 2 =−ν, will not differ by an integer, and we may obtain two linearly

independent solutions in the form of Frobenius series. Special considerations do

arise, however, whenν=m/2form=1, 3 , 5 ,...,andσ 1 −σ 2 =2ν=mis an

(odd positive) integer. When this happens, we may always obtain a solution in