Update

The code presented here has now been merged into BBN-Q's Mathieu function library and is accessible via Julia lang's general registry.

Mathieu-Functions

This is a Julia language library for computing periodic Mathieu functions with arbitrary q and n values. Mathieu functions are a class of very interesting and useful, but incredibly finnicky equations. They come about as the solutions to the following differential equation:

$$\frac{d^2}{dx^{2}} \psi(x) +(a-2q\cos(2x))\psi(x)=0$$

The specific problem that this package deals with is as follows: I need to plot and use periodic Mathieu functions with unusually large $q$ values. Unusually large here means $q$ values for which an advanced mathematics software like Mathematica, which is what comes in handy in doing high level formal mathematics in DFT, quantum mechanics and physics, cannot plot this function.

The word periodic is also key here, as Mathieu equation admits non-periodic solutions that I am not interested in at this point for my study. I might add them here as well in the future, but the periodic eigenfunctions are quite enough at this stage.

In most cases, we deal with small $q$ values and calculations with whatever computational software you use can proceed quite smoothly. But for a particular research paper, I need these functions for large negative and positive $q$ values, which comes about in the analysis of a real life quantum system.

(If the related paper is published, I will update the read me files to include a reference to it as well, to those who are interested.)

I have rigged a simple Julia package to calculate characteristic functions and periodic Mathieu functions. It is blazingly fast and therefore, it allows expansion of other periodic functions in terms of these by allowing fast numerical integrations of these functions. Something that is very tricky again for large negative or positive $q$ values in Mathematica. It can, of course, be used for all values of the $q$ parameter with the added benefit of beating Mathematica in plotting them for $q$ values that stymies Mathematica.

After writing these functions, I realized that there are some other Julia libraries already available that calculate the characteristic values for difference $q$ values as well. Please be sure to check them out as well:

• BBN-Q's Mathieu function library
• jebej's Mathieu function library

My calculation follows the brilliant textbook of Morse & Feshbach and the Digital Library of Mathematical Functions. The great efficiency of the Julia language gave rise to a pretty handy tool here that helped me greatly in my own calculations. I hope it can aid you in your projects as well!

As time goes on, I will pour in more technical details about the project and how the whole thing was developed in the end. It is a work in progress. So be sure to check in from time to time and leave a star on the repo, if you have found this useful.