Mathieu functions

The -periodic solutions of the Mathieu equation

which exist only when the point in the parameter plane lies on the boundary of the stability zones. A Mathieu function is even or odd, and is unique up to a factor; the second linearly-independent solution grows linearly in for , provided . The even Mathieu functions are the eigen functions of the integral equation

An analogous equation is satisfied by the odd Mathieu functions. The notation for Mathieu functions is:

For these functions reduce to the trigonometric system

and they possess the same orthogonality properties on the interval . The Mathieu functions admit Fourier-series expansions which converge for small ; the coefficients of these series are convergent power series in , for example,

References

[1]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2
[2]	H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955)
[3]	G. Sansone, "Equazioni differenziali nel campo reale" , 1 , Zanichelli (1948)
[4]	M.J.O. Strett, "Lamésche-, Mathieusche- und verwandte Funktionen in Physik und Technik" , Springer (1932)
[5]	N.W. Mac-Lachlan, "Theory and application of Mathieu functions" , Clarendon Press (1947)