Mathieu functions

The $ 2 \pi $- periodic solutions of the Mathieu equation

$$ \frac{d ^ {2} u }{dz ^ {2} } + ( a + 16q \cos 2z) u = 0,\ \ z \in \mathbf R , $$

which exist only when the point $ ( a, q) $ 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 $ z $ for $ | z | \rightarrow \infty $, provided $ q \neq 0 $. The even Mathieu functions are the eigen functions of the integral equation

$$ G( z) = \lambda \int\limits _ {- \pi } ^ \pi e ^ {k \cos z \cos t } G( t) dt,\ \ k = \sqrt 32q . $$

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

$$ ce _ {0} ( z, q), ce _ {1} ( z, q) , . . . ; \ se _ {1} ( z, q) , se _ {2} ( z, q) , . . . . $$

For $ q \rightarrow 0 $ these functions reduce to the trigonometric system

$$ 1 , \cos z , . . . ; \ \ \sin z, \sin 2z \dots $$

and they possess the same orthogonality properties on the interval $ (- \pi , \pi ) $. The Mathieu functions admit Fourier-series expansions which converge for small $ | q | \leq r _ {n} $; the coefficients of these series are convergent power series in $ q $, for example,

$$ ce _ {0} ( z, q) = \ 1 + $$

$$ + \sum _{n=1} ^ \infty \left [ 2 ^ {n+1} \frac{q ^ {n} }{( n!) ^ {2} } - \frac{n( 3n+ 4) 2 ^ {n+3} q ^ {n+2} }{(( n+ 1)!) ^ {2} } + O( q ^ {n+4} ) \right ] \cos 2nz. $$

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)