Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

the form of a Frobenius series corresponding to the larger root,σ 1 =ν=m/2,

as described above. However, for the smaller root,σ 2 =−ν=−m/2, we must

determine whether a second Frobenius series solution is possible by examining

the recurrence relation (18.78), which reads

n(n−m)an+an− 2 =0 forn≥ 2.

Sincemis anoddpositive integer in this case, we can use this recurrence relation

(starting witha 0 = 0) to calculatea 2 ,a 4 ,a 6 ,...in the knowledge that all these

terms will remain finite. It is possible in this case, therefore, to find a second

solution in the form of a Frobenius series, one that corresponds to the smaller

rootσ 2.

Thus, in general, for non-integerνwe have from (18.77) and (18.78)

an = −

1

n(n± 2 ν)

an− 2 forn=2, 4 , 6 ,...,

=0 forn=1, 3 , 5 ,....

Settinga 0 = 1 in each case, we obtain the two solutions

y±ν(x)=x±ν

[

1 −

x^2

2(2± 2 ν)

+

x^4

2 ×4(2± 2 ν)(4± 2 ν)

−···

]

.

It is customary, however, to set

a 0 =

1

2 ±νΓ(1±ν)

,

where Γ(x)isthegamma function, described in subsection 18.12.1; it may be

regarded as the generalisation of the factorial function to non-integer and/or

negative arguments.§The two solutions of (18.73) are then written asJν(x)and

J−ν(x), where

Jν(x)=

1

Γ(ν+1)

(x

2

)ν[

1 −

1

ν+1

(x

2

) 2

+

1

(ν+1)(ν+2)

1

2!

(x

2

) 4

−···

]

=

∑∞

n=0

(−1)n

n!Γ(ν+n+1)

(x

2

)ν+2n

; (18.79)

replacingνby−νgivesJ−ν(x). The functionsJν(x)andJ−ν(x) are calledBessel

functions of the first kind, of orderν. Since the first term of each series is a

finite non-zero multiple ofxνandx−ν, respectively, ifνis not an integer then

Jν(x)andJ−ν(x) are linearly independent. This may be confirmed by calculating

the Wronskian of these two functions. Therefore, for non-integerνthe general

solution of Bessel’s equation (18.73) is given by

y(x)=c 1 Jν(x)+c 2 J−ν(x). (18.80)

§In particular, Γ(n+1)=n!forn=0, 1 , 2 ,. ..,and Γ(n) is infinite ifnis any integer≤0.