Numerische Mathematik

, Volume 124, Issue 4, pp 723–752 | Cite as

Computing the complex zeros of special functions

Article

Abstract

The complex zeros of solutions of second order linear ODEs lie over certain curves in the complex plane called anti-Stokes lines. We consider the Liouville-Green (WKB) approximation for linear homogeneous second order ODEs \(w^{\prime \prime }(z)+A(z)w(z)=0\) with \(A(z)\) meromorphic, and describe the qualitative properties of the approximate anti-Stokes lines. Based on this qualitative description, we construct a fourth order method which efficiently computes the complex zeros of solutions of second order ODEs following the path of the approximate anti-Stokes lines. We illustrate the method with the computation of the zeros of parabolic cylinder functions, Bessel functions and generalized Bessel polynomials and describe specific algorithms for the computation of these zeros.

Mathematics Subject Classification (2000)

33F05 65H05 34M10 34C10 30E99 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain

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