Here, and throughout the remainder of §§10.47–10.60, 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 .)
and are the spherical Bessel functions of the first and second kinds, respectively; and are the spherical Bessel functions of the third kind.
, , 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 .
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.