# Mathieu function

In mathematics, the Mathieu functions are certain special functions useful for treating a variety of problems in applied mathematics, including:

They were introduced by Émile Léonard Mathieu (1868) in the context of the first problem.

## Mathieu equation

The canonical form for Mathieu's differential equation is[2]

${\displaystyle {\frac {d^{2}y}{dx^{2}}}+[a-2q\cos(2x)]y=0,}$ or with ${\displaystyle q=h^{2}:\;\;{\frac {d^{2}y}{dx^{2}}}+[a-2h^{2}\cos(2x)]y=0.}$

The Mathieu equation is a Hill equation with only 1 harmonic mode.

Closely related is Mathieu's modified differential equation

${\displaystyle {\frac {d^{2}y}{du^{2}}}-[a-2q\cosh(2u)]y=0}$

which follows on substitution ${\displaystyle u=ix}$.

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.[1] This is why they are also known as angular and radial Mathieu equation, respectively.

The substitution ${\displaystyle t=\cos(x)}$ transforms Mathieu's equation to the algebraic form

${\displaystyle (1-t^{2}){\frac {d^{2}y}{dt^{2}}}-t\,{\frac {dy}{dt}}+(a+2q(1-2t^{2}))\,y=0.}$

This has two regular singularities at ${\displaystyle t=-1,1}$ 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.

## Floquet solution

According to Floquet's theorem[3] (or Bloch's theorem),[4] for fixed values of a,q, Mathieu's equation admits a complex valued solution of form

${\displaystyle F(a,q,x)=\exp(i\mu \,x)\,P(a,q,x)}$

where ${\displaystyle \mu }$ is a complex number, the Floquet exponent (or sometimes Mathieu exponent), and P is a complex valued function which is periodic in ${\displaystyle x}$ with period ${\displaystyle \pi }$. However, P is in general not sinusoidal. In the example plotted below, ${\displaystyle a=1,\,q={\frac {1}{5}},\,\mu \approx 1+0.0995i}$ (real part, red; imaginary part, green):

## Mathieu sine and cosine

For fixed a,q, the Mathieu cosine ${\displaystyle C(a,q,x)}$ is a function of ${\displaystyle x}$ defined as the unique solution of the Mathieu equation which

1. takes the value ${\displaystyle C(a,q,0)=1}$,
2. is an even function, hence ${\displaystyle C^{\prime }(a,q,0)=0}$.

Similarly, the Mathieu sine ${\displaystyle S(a,q,x)}$ is the unique solution which

1. takes the value ${\displaystyle S^{\prime }(a,q,0)=1}$,
2. is an odd function, hence ${\displaystyle S(a,q,0)=0}$.

These are real-valued functions which are closely related to the Floquet solution:

${\displaystyle C(a,q,x)={\frac {F(a,q,x)+F(a,q,-x)}{2F(a,q,0)}}}$
${\displaystyle S(a,q,x)={\frac {F(a,q,x)-F(a,q,-x)}{2F^{\prime }(a,q,0)}}.}$

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

${\displaystyle C(a,0,x)=\cos({\sqrt {a}}x),\;S(a,0,x)={\frac {\sin({\sqrt {a}}x)}{\sqrt {a}}},}$

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

${\displaystyle C(a,q,x)\approx \cos({\sqrt {a}}x),\;\;S(a,q,x)\approx {\frac {\sin({\sqrt {a}}x)}{\sqrt {a}}}.}$

For example:

Red: C(0.3,0.1,x).
Red: C'(0.3,0.1,x).

## Periodic solutions

Given ${\displaystyle q}$, for countably many special values of ${\displaystyle a}$, called characteristic values or simply eigenvalues, the Mathieu equation admits solutions which are periodic with period ${\displaystyle 2\pi }$ or period ${\displaystyle \pi }$. The characteristic values of the Mathieu cosine and sine functions respectively are written ${\displaystyle a_{2n}(q),a_{2n+1}(q)}$ and ${\displaystyle b_{2n+1}(q),b_{2n+2}(q)}$, where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written ${\displaystyle CE(n,q,x)}$, or ${\displaystyle ce_{2n=q_{0}-1},ce_{2n+1=q_{0}},}$ and ${\displaystyle SE(n,q,x)}$, or ${\displaystyle se_{2n+1=q_{0}},se_{2n+2=q_{0}+1},}$ respectively where ${\displaystyle q_{0}}$ is an odd integer, although they are traditionally given a different normalization (namely, that their L2 norm equal ${\displaystyle \pi }$). Therefore, for positive q, we have

${\displaystyle C\left(a_{n}(q),q,x\right)={\frac {CE(n,q,x)}{CE(n,q,0)}}}$
${\displaystyle S\left(b_{n}(q),q,x\right)={\frac {SE(n,q,x)}{SE^{\prime }(n,q,0)}}.}$

Here are the first few periodic Mathieu cosine functions for q = 1:

Note that, for example, ${\displaystyle CE(1,1,x)}$ (green) resembles a cosine function, but with flatter hills and shallower valleys.

The series expansions of periodic Mathieu functions in ascending powers of ${\displaystyle q}$ are most easily obtained by perturbation theory.[5] For large values of ${\displaystyle q}$ two adjacent periodic solutions merge together to an asymptotic solution for large values of ${\displaystyle q}$ as shown in the book of Müller-Kirsten.

## Asymptotic solutions

For large values of ${\displaystyle q\equiv h^{2}}$ asymptotic expansions of periodic Mathieu functions have been given by R.B. Dingle and H.J.W. Müller.[6] Expansions for both small values of ${\displaystyle q}$ and large values of ${\displaystyle q}$ and their merging together have been given in.[7] For higher and higher barriers of the periodic potential ≈ ${\displaystyle q\cos(2x)}$, and that means for large values of ${\displaystyle q}$, the potential approaches asymptotically a system of independent harmonic oscillators with oscillator quantum number (say) ${\displaystyle \nu =n}$ for ${\displaystyle n}$ an integer or zero, and this leads to the leading approximation of the eigenvalue for ${\displaystyle q\rightarrow \infty }$. Higher order terms can be obtained by perturbation theory. One obtains in this way three pairs of asymptotic solutions which can be matched in common domains of overlap. Imposing the boundary conditions required for periodic solutions, one obtains the asymptotic expansion of the parameter ${\displaystyle \nu }$, i.e. ${\displaystyle \nu =n+O(1/q),}$ where ${\displaystyle n}$ is a positive integer or zero. Thus, for ${\displaystyle q\rightarrow \infty }$ the barriers of the periodic potential are infinitely high and there is no quantum mechanical tunneling through these barriers. Lowering these barriers, tunneling becomes possible, and the parameter ${\displaystyle \nu }$ is only approximately an integer. For details see the above references. In the study of asymptotic expansions, the behavior of the function under consideration for large values of a parameter like ${\displaystyle q}$ above plays a significant role. In the case of periodic Mathieu functions and spheroidal wave functions the large order behavior of the eigenvalue expansion has been derived in.[8]

## Solutions to the modified Mathieu equation

An important application of the modified Mathieu equation arises in the quantum mechanics of singular potentials. For the one particular singular potential ${\displaystyle V(r)={\frac {g^{2}}{r^{4}}}}$ the radial Schrödinger equation

${\displaystyle {\frac {d^{2}y}{dr^{2}}}+[k^{2}-{\frac {l(l+1)}{r^{2}}}-{\frac {g^{2}}{r^{4}}}]y=0}$

can be converted into the equation

${\displaystyle {\frac {d^{2}\phi }{dz^{2}}}+[2h^{2}\cosh 2z-{\big (}l+{\frac {1}{2}}{\big )}^{2}]\phi =0.}$

The transformation is achieved with the following substitutions

${\displaystyle y=r^{1/2}\phi ,r=\gamma e^{z},\gamma ={\frac {ig}{h}},h^{2}=ikg,h=e^{I\pi /4}(kg)^{1/2}.}$

With the replacements

${\displaystyle z\rightarrow iz,(l+{\frac {1}{2}})^{2}\rightarrow \lambda ,}$

one obtains the periodic Mathieu equation

${\displaystyle {\frac {d^{2}\phi }{dz^{2}}}+[\lambda -2h^{2}\cos 2z]\phi =0.}$

The solution of the above radial Schrödinger equation (for the specific singular potential) in terms of solutions of the modified Mathieu equation and the derivation of the S-matrix and the absorptivity has been given by Müller-Kirsten.[9]