Regular singular point

In mathematics, in the theory of ordinary differential equations in the complex plane

C

{\displaystyle \mathbb {C} }

, the points of

C

{\displaystyle \mathbb {C} }

are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.

Formal definitions

More precisely, consider an ordinary linear differential equation of n-th order

f

(

n

)

(

z

)

+

∑

i

=

0

n

−

1

p

i

(

z

)

f

(

i

)

(

z

)

=

0

{\displaystyle f^{(n)}(z)+\sum _{i=0}^{n-1}p_{i}(z)f^{(i)}(z)=0}

with pi(z) meromorphic functions.

The equation should be studied on the Riemann sphere to include the point at infinity as a possible singular point. A Möbius transformation may be applied to move ∞ into the finite part of the complex plane if required, see example on Bessel differential equation below.

Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are power series times complex powers (z − a)r near any given a in the complex plane where r need not be an integer; this function may exist, therefore, only thanks to a branch cut extending out from a, or on a Riemann surface of some punctured disc around a. This presents no difficulty for a an ordinary point (Lazarus Fuchs 1866). When a is a regular singular point, which by definition means that

p

n

−

i

(

z

)

{\displaystyle p_{n-i}(z)}

has a pole of order at most i at a, the Frobenius method also can be made to work and provide n independent solutions near a.

Otherwise the point a is an irregular singularity.