Mathieu function periodicity problem

Question

Bug introduced in 9.0 or earlier and fixed in 10.3.0

This is a documentation mistake in MathieuCharacteristicA, MathieuCharacteristicB, and MathieuCharacteristicExponent.

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According to the documentation, the Mathieu characteristic function generates parameter a:

The characteristic value Subscript[a, r] gives the value of the parameter a in y′′+(a-2q cos(2z))y=0 for which the solution has the form e^(i r z) f(z), where f(z) is an even function of z with period 2π.

However, I get the function f that are periodic of π instead of 2π. Here is the construction of the periodic function f (followed from Input 76 on page 1105 of The Mathematica Guidebook for Symbolics):

f[k_, q_, z_] := (MathieuC[MathieuCharacteristicA[k, q], q, z] + I Sign[k] MathieuS[MathieuCharacteristicB[k, q], q, z])/Exp[I k z]
Plot[{Abs@f[3, -1, z], Abs@f[1/3, -1, z]}, {z, -2 π, 2 π}, Axes -> False, Frame -> True, GridLines -> {π/2 Range[-3, 3, 2], {}}]

So why does the periodic function f have period of π instead of 2π ?

Answer 1

Actually, the characteristic value $a_r$ gives the value of the parameter $a$ in $y′′+(a-2q \cos(2z))\,y=0$ for which the solution has the form $e^{i r z} f(z)$, where $f(z)$ is an even function of $z$ with period $\pi$, not $2\pi$ as stated in the documentation.

See, for instance, Wikipedia, MathWorld, or the Digitial Library of Mathematical Functions.

The documentation is wrong.