# §15.11 Riemann’s Differential Equation

## §15.11(i) Equations with Three Singularities

The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). The most general form is given by

with

Here , , are the exponent pairs at the points , , , respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice for exponent pairs. Also, if any of , , , is at infinity, then we take the corresponding limit in (15.11.1).

The complete set of solutions of (15.11.1) is denoted by Riemann’s -symbol:

15.11.3

In particular,

15.11.4

denotes the set of solutions of (15.10.1).

## §15.11(ii) Transformation Formulas

A conformal mapping of the extended complex plane onto itself has the form

where , , , are real or complex constants such that . These constants can be chosen to map any two sets of three distinct points and onto each other. Symbolically:

15.11.6

The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by

15.11.7

We also have

15.11.8

for arbitrary and .