32.2. SOME SIMPLE OBSERVATIONS ON POWER SERIES 615

32.2 Some Simple Observations on Power SeriesThis section is a review of a few facts about power series which should have been learnedin calculus. If you have not seen these things, which may well be the case given the waycalculus courses are systematically watered down, see my book Calculus of one and manyvariables on my web page or my differential equations book [25].

Definition 32.2.1 A function f is analytic in some open set U if for each a ∈U, f (x) =∑

∞k=0 ak (x−a)k for all x close enough to a. In other words, you can get the function near a

by a power series.

Theorem 32.2.2 Suppose f (x) = ∑∞n=0 an (x−a)n for x near a and suppose a0 ̸= 0. Then

f (x)−1 =1a0

+h(x)

where h(x) = ∑∞n=1 bn (x−a)n so h(a) = 0.

Proof: It turns out that f (x)−1 has a power series representation near a and so f (a)−1 =1/a0. ■

Theorem 32.2.3 Suppose f (x) = ∑∞n=0 anxn and g(x) = ∑

∞n=0 bnxn for x near 0. Then

f (x)g(x) also has a power series near 0 and in fact,

f (x)g(x) =∞

∑n=0

(n

∑k=0

an−kbk

)xn. (32.3)

Proof: See the material on power series in my calculus book. However, it is quiteplausible. (

∞

∑n=0

anxn

)(∞

∑n=0

bnxn

)=

(a0 +a1x+a2x2 + · · ·

)(b0 +b1x+b2x2 + · · ·

)Now formally multiply the two power series like they were polynomials and collect terms.This will yield 32.3. ■

32.3 Regular Singular PointsFirst of all, here is the definition of what a regular singular point is.

Definition 32.3.1 A differential equation has a regular singular point at 0 if the equationcan be written in the form

x2y′′+ xb(x)y′+ c(x)y = 0 (32.4)

where

b(x) =∞

∑n=0

bnxn,∞

∑n=0

cnxn = c(x)