# Radial Mathieu functions, divergence problem

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

• Please read the wiki info for the tag bugs. Its use is reserved. Jan 12, 2020 at 22:27
• 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. Jan 12, 2020 at 22:30
• Jan 12, 2020 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}. Jan 13, 2020 at 0:00
• 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. Mar 31, 2020 at 7:18