# §10.2 Definitions

## §10.2(i) Bessel’s Equation

This differential equation has a regular singularity at with indices , and an irregular singularity at of rank 1; compare §§2.7(i) and 2.7(ii).

## §10.2(ii) Standard Solutions

### ¶ Bessel Function of the First Kind

10.2.2

This solution of (10.2.1) is an analytic function of , except for a branch point at when is not an integer. The principal branch of corresponds to the principal value of 4.2(iv)) and is analytic in the -plane cut along the interval .

When , is entire in .

For fixed each branch of is entire in .

### ¶ Bessel Function of the Second Kind (Weber’s Function)

When is an integer the right-hand side is replaced by its limiting value:

Whether or not is an integer has a branch point at . The principal branch corresponds to the principal branches of in (10.2.3) and (10.2.4), with a cut in the -plane along the interval .

Except in the case of , the principal branches of and are two-valued and discontinuous on the cut ; compare §4.2(i).

Both and are real when is real and .

For fixed each branch of is entire in .

### ¶ Bessel Functions of the Third Kind (Hankel Functions)

These solutions of (10.2.1) are denoted by and , and their defining properties are given by

10.2.5

as in , and

10.2.6

as in , where is an arbitrary small positive constant. Each solution has a branch point at for all . The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the -plane along the interval .

The principal branches of and are two-valued and discontinuous on the cut .

For fixed each branch of and is entire in .

### ¶ Branch Conventions

Except where indicated otherwise, it is assumed throughout the DLMF that the symbols , , , and denote the principal values of these functions.

### ¶ Cylinder Functions

The notation denotes , , , , or any nontrivial linear combination of these functions, the coefficients in which are independent of and .

## §10.2(iii) Numerically Satisfactory Pairs of Solutions

Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case . When , is replaced by throughout.

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.
Pair Interval or Region
neighborhood of 0 in
neighborhood of in