Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

singular points, whereas any singular point not satisfying both these criteria is

termed anirregularoressentialsingularity.

Legendre’s equation has the form

(1−z^2 )y′′− 2 zy′+(+1)y=0, (16.8)

whereis a constant. Show thatz=0is an ordinary point andz=± 1 are regular singular

points of this equation.

Firstly, divide through by 1−z^2 to put the equation into our standard form (16.7):

y′′−

2 z

1 −z^2

y′+

(+1)

1 −z^2

y=0.

Comparing this with (16.7), we identifyp(z)andq(z)as

p(z)=

− 2 z

1 −z^2

=

− 2 z

(1 +z)(1−z)

,q(z)=

(+1)

1 −z^2

=

(+1)

(1 +z)(1−z)

.

By inspection,p(z)andq(z) are analytic atz= 0, which is therefore an ordinary point,
but both diverge forz=±1, which are thus singular points. However, atz=1wesee
that both (z−1)p(z)and(z−1)^2 q(z) are analytic and hencez= 1 is a regular singular
point. Similarly, atz=−1 both (z+1)p(z)and(z+1)^2 q(z) are analytic, and it too is a
regular singular point.

So far we have assumed thatz 0 is finite. However, we may sometimes wish to

determine the nature of the point|z|→∞. This may be achieved straightforwardly

by substitutingw=1/zinto the equation and investigating the behaviour at

w=0.

Show that Legendre’s equation has a regular singularity at|z|→∞.

Lettingw=1/z, the derivatives with respect tozbecome

dy

dz

=

dy

dw

dw

dz

=−

1

z^2

dy

dw

=−w^2

dy

dw

,

d^2 y

dz^2

=

dw

dz

d

dw

(

dy

dz

)

=−w^2

(

− 2 w

dy

dw

−w^2

d^2 y

dw^2

)

=w^3

(

2

dy

dw

+w

d^2 y

dw^2

)

.

If we substitute these derivatives intoLegendre’s equation (16.8) we obtain
(
1 −

1

w^2

)

w^3

(

2

dy

dw

+w

d^2 y

dw^2

)

+2

1

w

w^2

dy

dw

+(+1)y=0,

which simplifies to give

w^2 (w^2 −1)

d^2 y

dw^2

+2w^3

dy

dw

+(+1)y=0.

Dividing through byw^2 (w^2 −1) to put the equation into standard form, and comparing
with (16.7), we identifyp(w)andq(w)as

p(w)=

2 w

w^2 − 1

,q(w)=

(+1)

w^2 (w^2 −1)

.

Atw=0,p(w) is analytic butq(w) diverges, and so the point|z|→∞is a singular point
of Legendre’s equation. However, sincewpandw^2 qare both analytic atw=0,|z|→∞
is a regular singular point.