  We describe a variety of methods to compute the functions $K_{ia}(x)$,
$L_{ia}(x)$ and their derivatives for real $a$ and positive $x$. These
functions are numerically satisfactory independent solutions of the
differential equation $x^2 w'' +x w' +(a^2 -x^2)w=0$. In an accompanying paper
(Algorithm xxx: Modified Bessel functions of imaginary order and positive
argument) we describe the implementation of these methods in Fortran 77 codes.
