Mathieu function rescale problem

Question

The Mathieu functions are the solutions for the equation

$$ y''+(a-2q\cos(2z))y=0 $$

If we require the solution has the form

$$ y(z) = e^{i r z}f(z) $$

where $f(z)$ is a periodic function with period of $2\pi$, then the parameter $a$ should satisfy Mathieu characteristic function

$$ a=\text{MatheiuCharacteristicA}(r,q) $$

My question is what's the condition of $a$ should satisfy, if the solution can be written as

$$ y(z)=e^{irz}f(z) $$

where $f(z)$ is a periodic function with period 1, i.e., $f(z+1)=f(z)$?

Update

It seems that the period of the original Mathieu function should be $\pi$ instead of $2\pi$, according to Wikipedia, MathWorld, and DLMF, and the documentation of Mathematica may be incorrect on that. That means the ordinary scaling of the equation works fine.

Answer 1

I recommend you to use Scilab Mathieu Functions Toolbox, not Mathematica.

The parameter a (or b) is a function of order r and value of q - $a_r(q)$ (or $b_r(q)$), the dependency may be illustrated as follows (from aforementioned toolbox).

You can read the following documents about Mathieu functions:

• N.W. McLachlan, Theory and Application of Mathieu Functions, Oxford Univ. Press, 1947.
• Chapter 28 Mathieu Functions and Hill's Equation. Digital Library of Mathematical Functions. NIST.
• help and documents of Scilab Mathieu Functions Toolbox

If you are interested - the toolbox installation procedure is as follows:

1. Install Scilab
2. Start Scilab
3. Update ATOMS database with atomsSystemUpdate
4. Install Mathieu Functions Toolbox with atomsInstall('Mathieu')
5. Restart Scilab, launch its help (? -> Scilab Help), navigate to 'Mathieu functions' or launch demo (? -> Scilab Demonstrations -> Mathieu functions).