§13.2 Definitions and Basic Properties

§13.2(i) Differential Equation

¶ Kummer’s Equation

This equation has a regular singularity at the origin with indices 0 and , and an irregular singularity at infinity of rank one. It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing by , letting , and subsequently replacing the symbol by . In effect, the regular singularities of the hypergeometric differential equation at and coalesce into an irregular singularity at .

¶ Standard Solutions

The first two standard solutions are:

13.2.2

and

13.2.3

except that does not exist when is a nonpositive integer. In other cases

The series (13.2.2) and (13.2.3) converge for all . is entire in and , and is a meromorphic function of . is entire in , , and .

Although does not exist when , , many formulas containing continue to apply in their limiting form. In particular,

When , , is a polynomial in of degree not exceeding ; this is also true of provided that is not a nonpositive integer.

Another standard solution of (13.2.1) is , which is determined uniquely by the property

13.2.6, ,

where is an arbitrary small positive constant. In general, has a branch point at . The principal branch corresponds to the principal value of in (13.2.6), and has a cut in the -plane along the interval ; compare §4.2(i).

When , , the following equation can be combined with (13.2.9) and (13.2.10):

§13.2(ii) Analytic Continuation

Except when each branch of is entire in and . Unless specified otherwise, however, is assumed to have its principal value.

§13.2(iii) Limiting Forms as

Next, in cases when or , where is a nonnegative integer,

In all other cases

13.2.16, ,

§13.2(iv) Limiting Forms as

Except when (polynomial cases),

where is an arbitrary small positive constant.

For see (13.2.6).

§13.2(v) Numerically Satisfactory Solutions

Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are

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