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

Amparo Gil, Javier Segura, Nico M. Temme

TL;DR

A variety of methods to compute the functions K, L, IA and X and their derivatives for real and positive real and negative positive values and the implementation in Fortran 77 codes is described.

Abstract

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.

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

TL;DR

A variety of methods to compute the functions K, L, IA and X and their derivatives for real and positive real and negative positive values and the implementation in Fortran 77 codes is described.

Abstract

We describe a variety of methods to compute the functions , and their derivatives for real and positive . These functions are numerically satisfactory independent solutions of the differential equation . 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.

Paper Structure

This paper contains 11 sections, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Methods of computation
  3. Series expansions
  4. Asymptotic expansions for large $x$
  5. Airy-type uniform asymptotic expansions
  6. Non-oscillating integral representations
  7. Monotonic case ($x>a$)
  8. Oscillatory case ($x<a$)
  9. Quadrature rule
  10. Continued fraction method
  11. Range of computation and scaled functions