# §10.47 Definitions and Basic Properties

## §10.47(i) Differential Equations

Here, and throughout the remainder of §§10.4710.60, is a nonnegative integer. (This is in contrast to other treatments of spherical Bessel functions, including Abramowitz and Stegun (1964, Chapter 10), in which can be any integer. However, there is a gain in symmetry, without any loss of generality in applications, on restricting .)

Equations (10.47.1) and (10.47.2) each have a regular singularity at with indices , , and an irregular singularity at of rank 1; compare §§2.7(i)2.7(ii).

## §10.47(ii) Standard Solutions

### ¶ Equation (10.47.1)

10.47.4
10.47.5
10.47.6

and are the spherical Bessel functions of the first and second kinds, respectively; and are the spherical Bessel functions of the third kind.

### ¶ Equation (10.47.2)

10.47.7
10.47.8
10.47.9

, , and are the modified spherical Bessel functions.

Many properties of , , , , , , and follow straightforwardly from the above definitions and results given in preceding sections of this chapter. For example, , , , , , , and are all entire functions of .

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

For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols , , , and replaced by , , , and , respectively.

For (10.47.2) numerically satisfactory pairs of solutions are and in the right half of the -plane, and and in the left half of the -plane.