# Mathieu equation

The following ordinary differential equation with real coefficients:

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

It was introduced by E. Mathieu  in the investigation of the oscillations of an elliptic membrane; it is a particular case of a Hill equation.

A fundamental system of solutions of the Mathieu equation has the form

$$\tag{* } u _ {1} ( z) = e ^ {\alpha z } \phi ( z),\ \ u _ {2} ( z) = u _ {1} (- z) ,$$

for $\alpha \neq ni$, $n$ an integer, where $\phi ( z)$ is a $\pi$- periodic function and the Lyapunov characteristic exponent $\alpha$ is either real or purely imaginary. For $\mathop{\rm Im} \alpha = 0$ one of the solutions grows unboundedly, whereas the other tends to zero as $z \rightarrow + \infty$( instability zones in the plane of the parameters $a , b$); for $\mathop{\rm Re} \alpha = 0$ these solutions are both bounded (stability zones). On the boundary of these zones (the case excluded in (*)) one of the functions of the fundamental system of solutions is either $\pi$- periodic or $2 \pi$- periodic (the latter is called a Mathieu function, cf. Mathieu functions), while the second is obtained from the first through multiplication by $z$. The instability zones have the form of curvilinear triangles with vertices at the points $a = n ^ {2}$, $b = 0$, $n = 0, 1 ,\dots$( see , ).

The Mathieu equation is known also in a different form (see ).

How to Cite This Entry:
Mathieu equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mathieu_equation&oldid=47790
This article was adapted from an original article by V.M. Starzhinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article