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