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I am working on a project that requires the utilization of even Mathieu functions.

This is the definition of my functions:

  1. Even Mathieu function: Ce[a, q, z] = MathieuC[a, q, z]
  2. Characteristic number of order m: am[q] = MathieuCharacteristicA[m, q]

  3. Radial Mathiew function: 

    Cem[q,x] = MathieuC[MathieuCharacteristicA[m, q], q, I x]

In my project, I have to work with the function Cem[q, x] with indexes m = 2 r, r=1, 2, 3, 4, 5, 6, 7, 8, ..., 0 <= q <= 400, and 0 < x < 20.

So, this is the problem: The function Cem[q, x] explodes (becomes very badly behaved) when it is at the range x > 2.5, and get really worse for larger values of q. The next figure shows what I mean. Could anyone give me a suggestion on how to solve this problem? T

enter image description here

Jorge Otalora
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  • Please read the wiki info for the tag [tag:bugs]. Its use is reserved. – Michael E2 Jan 12 '20 at 22:27
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    Try raising the WorkingPrecision: Plot[MathieuC[MathieuCharacteristicA[4, 10], 10, I x], {x, 0, 4}, WorkingPrecision -> 100] -- I think this question has been asked before. – Michael E2 Jan 12 '20 at 22:30
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    Related: https://mathematica.stackexchange.com/q/124993/4999, https://mathematica.stackexchange.com/q/83032/4999 – Michael E2 Jan 12 '20 at 22:32
  • Thank you for your help. Nevertheless, I followed your posts, and I still have some issues. How can I do to evaluate the Mathieu functions (for instance, MathieuC[MathieuCharacteristicA[4, 10], 10, I x]) for arbitrary larger values of x (for instance x>3) and avoiding the problems mentioned in my original post? For instance, by doing Plot[MathieuC[MathieuCharacteristicA[4, 10], 10, I x], {x, 4, 10}, WorkingPrecision -> 100], it fails. Accordingly, I am afraid that I would also have numerical errors by doing numerical Integrations of such Mathieu functions in a range of {x,0,10}. – Jorge Otalora Jan 13 '20 at 00:00
  • Basically, you're out of luck with Mathematica here. You may want to try GLS's implementation of Mathieu functions, importing them into Mathematica as I did for Bessel functions here. – Ruslan Mar 31 '20 at 07:18

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