Mathematical Methods for Physics and Engineering : A Comprehensive Guide

16.2 SERIES SOLUTIONS ABOUT AN ORDINARY POINT

Equation Regular Essential

singularities singularities

Hypergeometric

z(1−z)y′′+[c−(a+b+1)z]y′−aby=0 0 , 1 ,∞ —

Legendre

(1−z^2 )y′′− 2 zy′+(+1)y=0 − 1 , 1 ,∞ —

Associated Legendre

(1−z^2 )y′′− 2 zy′+

[

(+1)−

m^2

1 −z^2

]

y=0 − 1 , 1 ,∞ —

Chebyshev

(1−z^2 )y′′−zy′+ν^2 y=0 − 1 , 1 ,∞ —

Confluent hypergeometric

zy′′+(c−z)y′−ay=0 0 ∞

Bessel

z^2 y′′+zy′+(z^2 −ν^2 )y=0 0 ∞

Laguerre

zy′′+(1−z)y′+νy=0 0 ∞

Associated Laguerre

zy′′+(m+1−z)y′+(ν−m)y=0 0 ∞

Hermite

y′′− 2 zy′+2νy=0 — ∞

Simple harmonic oscillator

y′′+ω^2 y=0 — ∞

Table 16.1 Important second-order linear ODEs in the physical sciences and

engineering.

Table 16.1 lists the singular points of several second-order linear ODEs that

play important roles in the analysis of many problems in physics and engineering.

A full discussion of the solutions to each of the equations in table 16.1 and their

properties is left until chapter 18. We now discuss the general methods by which

series solutions may be obtained.

16.2 Series solutions about an ordinary point

Ifz=z 0 is an ordinary point of (16.7) then it may be shown thateverysolution

y(z) of the equation is also analytic atz=z 0. From now on we will takez 0 as the

origin, i.e.z 0 = 0. If this is not already the case, then a substitutionZ=z−z 0

will make it so. Since every solution is analytic,y(z) can be represented by a