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.
Read more about this topic: Mathieu Wavelet
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