Mathieu functions for purely imaginary parameters

  • Authors:

  • C. H. Ziener;M. RüCkl;T. Kampf;W. R. Bauer;H. P. Schlemmer

  • Affiliations:

  • German Cancer Research Center - DKFZ, Department of Radiology, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany and Julius-Maximilians-Universität Würzburg, Medizinische Klinik I, Ober ...;Humboldt-University Berlin, Department of Physics, Newtonstraíe 15, 12489 Berlin, Germany and Julius-Maximilians-Universität Würzburg, Experimentelle Physik 5, Am Hubland, 97074 W&# ...;Julius-Maximilians-Universität Würzburg, Experimentelle Physik 5, Am Hubland, 97074 Würzburg, Germany;Julius-Maximilians-Universität Würzburg, Medizinische Klinik I, Oberdürrbacher Straíe 6, 97080 Würzburg, Germany;German Cancer Research Center - DKFZ, Department of Radiology, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2012

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Abstract

For the Mathieu differential equation y^''(x)+[a-2qcos(x)]y(x)=0 with purely imaginary parameter q=is, the characteristic value a exhibits branching points. We analyze the properties of the Mathieu functions and their Fourier coefficients in the vicinity of the branching points. Symmetry relations for the Mathieu functions as well as the Fourier coefficients behind a branching point are given. A numerical method to compute Mathieu functions for all values of the parameter s is presented.