616 CHAPTER 32. SOLUTIONS NEAR A REGULAR SINGULAR POINT

for all x near 0. Such functions are called analytic in this section. More generally, a differ-ential equation

P(x)y′′+Q(x)y′+R(x)y = 0 (32.5)

where P,Q,R are analytic near a has a regular singular point at a if it can be written in theform

(x−a)2 y′′+(x−a)b(x)y′+ c(x)y = 0 (32.6)

where

b(x) =∞

∑n=0

bn (x−a)n ,∞

∑n=0

cn (x−a)n = c(x)

for all |x−a| small enough. The equation 32.5 has a singular point at a if P(a) = 0.

The following table emphasizes the similarities between the Euler equations and theregular singular point equations. I have featured the point 0. If you are interested in anotherpoint a, you just replace x with x−a everywhere it occurs.

Euler equation regular singular point

form of equation x2y′′+ xb0y′+ c0y = 0x2y′′+ x(b0 +b1x+ · · ·)y′

+(c0 + c1x+ · · ·)y = 0indicial equation r (r−1)+b0r+ c0 = 0 r (r−1)+b0r+ c0 = 0one solution y = xr y = xr

∑∞k=0 akxk, a0 = 1.

Recognizing Regular Singular Points

How do you know a singular differential equation can be written a certain way? Inparticular, how can you recognize a regular singular point when you see one? Suppose

P(x)y′′+Q(x)y′+R(x)y = 0

where all of P,Q,R are analytic functions near a. How can you tell if it has a regularsingular point at a? Here is how. It has a regular singular point at a if

limx→a

(x−a)Q(x)P(x)

exists

limx→a

(x−a)2 R(x)P(x)

exists

If these conditions hold, then by theorems in complex analysis it will be the case that

(x−a)Q(x)P(x)

=∞

∑n=0

bn (x−a)n ,

and

(x−a)2 R(x)P(x)

=∞

∑n=0

cn (x−a)n

for x near a. Indeed, equations of this form reduce to the form in 32.6 upon dividing byP(x) and multiplying by (x−a)2 .