Mathematical Methods for Physics and Engineering : A Comprehensive Guide

16.3 SERIES SOLUTIONS ABOUT A REGULAR SINGULAR POINT

then at least one ofp(z)andq(z) is not analytic atz= 0, and in general we

should not expect to find a power series solution of the form (16.9). We must

therefore extend the method to include a more general form for the solution. In

fact, it may be shown (Fuch’s theorem) that there existsat least onesolution to

the above equation, of the form

y=zσ

∑∞

n=0

anzn, (16.12)

where the exponentσis a number that may be real or complex and wherea 0 =0

(since, if it were otherwise,σcould be redefined asσ+1 orσ+2 or···so as to

makea 0 = 0). Such a series is called a generalised power series orFrobenius series.

As in the case of a simple power series solution, the radius of convergence of the

Frobenius series is, in general, equal to the distance to the nearest singularity of

the ODE.

Sincez= 0 is a regular singularity of the ODE, it follows thatzp(z)andz^2 q(z)

are analytic atz= 0, so that we may write

zp(z)≡s(z)=

∑∞

n=0

snzn,

z^2 q(z)≡t(z)=

∑∞

n=0

tnzn,

where we have defined the analytic functionss(z)andt(z) for later convenience.

The original ODE therefore becomes

y′′+

s(z)

z

y′+

t(z)

z^2

y=0.

Let us substitute the Frobenius series (16.12) into this equation. The derivatives

of (16.12) with respect tozare given by

y′=

∑∞

n=0

(n+σ)anzn+σ−^1 , (16.13)

y′′=

∑∞

n=0

(n+σ)(n+σ−1)anzn+σ−^2 , (16.14)

and we obtain

∑∞

n=0

(n+σ)(n+σ−1)anzn+σ−^2 +s(z)

∑∞

n=0

(n+σ)anzn+σ−^2 +t(z)

∑∞

n=0

anzn+σ−^2 =0.

Dividing this equation through byzσ−^2 , we find

∑∞

n=0

[(n+σ)(n+σ−1) +s(z)(n+σ)+t(z)]anzn=0. (16.15)