Mathieu Wavelet - Mathieu Functions: Cosine-elliptic and Sine-elliptic Functions

Mathieu Functions: Cosine-elliptic and Sine-elliptic Functions

In general, the solutions of Mathieu equation are not periodic. However, for a given q, periodic solutions exist for infinitely many special values (eigenvalues) of a. For several physically relevant solutions y must be periodic of period or . It is convenient to distinguish even and odd periodic solutions, which are termed Mathieu functions of first kind.

One of four simpler types can be considered: Periodic solution ( or ) symmetry (even or odd).

For, the only periodic solutions y corresponding to any characteristic value or have the following notations:

ce and se are abbreviations for cosine-elliptic and sine-elliptic, respectively.

• Even periodic solution:

• Odd periodic solution:

where the sums are taken over even (respectively odd) values of m if the period of y is (respectively ).

Given r, we denote henceforth by, for short.

Interesting relationships are found when, :

Figure 1 shows two illustrative waveform of elliptic cosines, whose shape strongly depends on the parameters and q.