# Mathieu functions

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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)
How to Cite This Entry:
Mathieu functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mathieu_functions&oldid=47791
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article