Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments

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 $x^2{w}^{\mathrm{\prime }\mathrm{\prime }}+x{w}^{\mathrm{\prime }}+(a^2-x^2)w=0$. 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.