# §13.14 Definitions and Basic Properties

## §13.14(i) Differential Equation

### ¶ Whittaker’s Equation

13.14.1

This equation is obtained from Kummer’s equation (13.2.1) via the substitutions , , and . It has a regular singularity at the origin with indices , and an irregular singularity at infinity of rank one.

### ¶ Standard Solutions

Standard solutions are:

13.14.2
13.14.3

except that does not exist when .

The series

13.14.6,

converge for all .

In general and are many-valued functions of with branch points at and . The principal branches correspond to the principal branches of the functions and on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i).

Although does not exist when , many formulas containing continue to apply in their limiting form. For example, if , then

If , where , then

## §13.14(ii) Analytic Continuation

Except when , each branch of the functions and is entire in and . Also, unless specified otherwise and are assumed to have their principal values.

## §13.14(v) Numerically Satisfactory Solutions

Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are

A fundamental pair of solutions that is numerically satisfactory in the sector near the origin is

When is an integer we may use the results of §13.2(v) with the substitutions , , and , where is the solution of (13.14.1) corresponding to the solution of (13.2.1).