# Root finding: zeroes of Mathieu function

I am finding the roots of the Mathieu sine function, and find Mathematica and Maple do not agree on the solutions.

For example, consider the solutions of Abs[MathieuS[4x, 4, Pi]] = 0, for 2 < x < 3.

Mathematica finds the solutions x=2.31536 and x=2.66776, whereas Maple only agrees with the solution x=2.31536.

I show plots from Mathematica (top) and Maple (bottom) illustrating the disagreement.  Why are the results inconsistent?

• You may want to try Plot[{Re@#, Im@#} &@MathieuS[4 x, 4, Pi], {x, 2, 3}, Evaluated -> True] – Dr. belisarius Nov 6 '14 at 0:21
• I guess their definition of Mathieu functions are different. For example, the Mathie characteristic functions in maple only take integer argument, but in Mathematica it takes real argument. I tried but never figure out the differences. – xslittlegrass Nov 6 '14 at 1:44
• for special functions, you need to make sure software uses the definition you expect, as for some special functions they can use slightly different definition. So you need to look up Maple definition of Mathieu and compare that to Mathematica to make sure which one you want to use. – Nasser Nov 6 '14 at 3:00

Plot[Abs[MathieuS[4 x, 4, Pi]/MathieuSPrime[4 x, 4, 0]], {x, 2, 3}] 