# 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) |

**How to Cite This Entry:**

Mathieu functions.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mathieu_functions&oldid=14316