# §10.25 Definitions

## §10.25(i) Modified Bessel’s Equation

This equation is obtained from Bessel’s equation (10.2.1) on replacing by , and it has the same kinds of singularities. Its solutions are called modified Bessel functions or Bessel functions of imaginary argument.

## §10.25(ii) Standard Solutions

10.25.2

This solution has properties analogous to those of , defined in §10.2(ii). In particular, the principal branch of is defined in a similar way: it corresponds to the principal value of , is analytic in , and two-valued and discontinuous on the cut .

The defining property of the second standard solution of (10.25.1) is

10.25.3

as in . It has a branch point at for all . The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in , and two-valued and discontinuous on the cut .

Both and are real when is real and .

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.

### ¶ Symbol

Corresponding to the symbol introduced in §10.2(ii), we sometimes use to denote , , or any nontrivial linear combination of these functions, the coefficients in which are independent of and .

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

Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). It is assumed that . When , is replaced by .

Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.
Pair Region