- vibrating elliptical drumheads,
- quadrupole mass analyzers and quadrupole ion traps for mass spectrometry
- wave motion in periodic media, such as ultracold atoms in an optical lattice
- the phenomenon of parametric resonance in forced oscillators,
- exact plane wave solutions in general relativity,
- the Stark effect for a rotating electric dipole,
- in general, the solution of differential equations that are separable in elliptic cylindrical coordinates.
They were introduced by Template:Harvs in the context of the first problem.
The canonical form for Mathieu's differential equation is
The Mathieu equation is a Hill equation with only 1 harmonic mode.
Closely related is Mathieu's modified differential equation
The two above equations can be obtained from the Helmholtz equation in two dimensions, by expressing it in elliptical coordinates and then separating the two variables. This is why they are also known as angular and radial Mathieu equation, respectively.
This has two regular singularities at and one irregular singularity at infinity, which implies that in general (unlike many other special functions), the solutions of Mathieu's equation cannot be expressed in terms of hypergeometric functions.
Mathieu's differential equations arise as models in many contexts, including the stability of railroad rails as trains drive over them, seasonally forced population dynamics, the four-dimensional wave equation, and the Floquet theory of the stability of limit cycles.
where is a complex number, the Mathieu exponent, and P is a complex valued function which is periodic in with period . However, P is in general not sinusoidal. In the example plotted below, (real part, red; imaginary part; green):
Mathieu sine and cosine
- takes the value ,
- is an even function, hence .
- takes the value ,
- is an odd function, hence .
These are real-valued functions which are closely related to the Floquet solution:
The general solution to the Mathieu equation (for fixed a,q) is a linear combination of the Mathieu cosine and Mathieu sine functions.
A noteworthy special case is
i.e. when the corresponding Helmholtz equation problem has circular symmetry.
In general, the Mathieu sine and cosine are aperiodic. Nonetheless, for small values of q, we have approximately
Given , for countably many special values of , called characteristic values, the Mathieu equation admits solutions which are periodic with period . The characteristic values of the Mathieu cosine, sine functions respectively are written , where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written respectively, although they are traditionally given a different normalization (namely, that their L2 norm equal ). Therefore, for positive q, we have
Here are the first few periodic Mathieu cosine functions for q = 1:
Solutions to the modified Mathieu equation
- Monochromatic electromagnetic plane wave, an example of an important exact plane wave solution to the Einstein field equation in general relativity which is expressed using Mathieu cosine functions.
- Inverted pendulum
- Lamé function
- Gertrude Blanch, "Chapter 20. Mathieu Functions", in Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972)
- Timothy Jones, Mathieu's Equations and the Ideal rf-Paul Trap (2006)
- Weisstein, Eric W., "Mathieu function", MathWorld.
- Mathieu equation, EqWorld
- List of equations and identities for Mathieu Functions functions.wolfram.com
- NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation