Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 29
Regular Singularities
The method of power series solutions discussed in chapter 28 fails when
the series is expanded about a singularity. In the special case of regular
singularity this problem can be rectified with method of Frobenius, which
we will discuss in chapter 30. We recall the definition of a regular singular
point here.
Definition 29.1. Let
a(t)y ′′ + b(t)y ′ + c(t)y = 0 (29.1)
have a singular point at t 0 , i.e., a(t 0 ) = 0. Then if the limits
(t − t 0 )b(t)
lim
= L 1 (29.2)
t→t 0 a(t)
(t − t 0 ) 2 c(t)
lim
= L 2 (29.3)
t→t 0 a(t)
exist and are finite we say that t 0 is a regular singularity of the differential
equation. Note that these limits will always exist when the
arguments of the limit are analytic (infinitely differentiable) at t 0 . If either
of the limits does not exist or are infinite (or the arguments of either limit
is not analytic) then we say that t 0 is an irregular singularity of the
differential equation. If a(t 0 ) ≠ 0 then we call t 0 an ordinary point of
the differential equation.
Theorem 29.2. The power series method of chapter 28 works whenever the
series is expanded about an ordinary point, and the radius of convergence
of the series is the distance from t 0 to the nearest singularity of a(t).
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