Mathieu function

In some usages, Mathieu function refers to solutions of the Mathieu differential equation for arbitrary values of $a$ and $q$. When no confusion can arise, other authors use the term to refer specifically to $\backslash pi$- or $2\backslash pi$-periodic solutions, which exist only for special values of $a$ and $q$.Arscott (1964), chapter III More precisely, for given (real) $q$ such periodic solutions exist for an infinite number of values of $a$, called characteristic numbers, conventionally indexed as two separate sequences $a\_n(q)$ and $b\_n(q)$, for $n\; =\; 1,\; 2,\; 3,\; \backslash ldots$. The corresponding functions are denoted $\backslash text\{ce\}\_n(x,\; q)$ and $\backslash text\{se\}\_n(x,\; q)$, respectively. They are sometimes also referred to as cosine-elliptic and sine-elliptic, or Mathieu functions of the first kind.

As a result of assuming that $q$ is real, both the characteristic numbers and associated functions are real-valued.Arscott (1964) 43–44

$\backslash text\{ce\}\_n(x,\; q)$ and $\backslash text\{se\}\_n(x,\; q)$ can be further classified by parity and periodicity (both with respect to $x$), as follows:

:

The indexing with the integer $n$, besides serving to arrange the characteristic numbers in ascending order, is convenient in that $\backslash text\{ce\}\_n(x,\; q)$ and $\backslash text\{se\}\_n(x,\; q)$ become proportional to $\backslash cos\; nx$ and $\backslash sin\; nx$ as $q\; \backslash rightarrow\; 0$. With $n$ being an integer, this gives rise to the classification of $\backslash text\{ce\}\_n$ and $\backslash text\{se\}\_n$ as Mathieu functions (of the first kind) of integral order. For general $a$ and $q$, solutions besides these can be defined, including Mathieu functions of fractional order as well as non-periodic solutions.

Closely related are the modified Mathieu functions, also known as radial Mathieu functions, which are solutions of Mathieu's modified differential equation

:$\backslash frac\{d^2y\}\{dx^2\}\; -\; (a\; -\; 2q\backslash cosh\; 2x)y\; =\; 0,$

which can be related to the original Mathieu equation by taking $x\; \backslash to\; \backslash pm\; \{\backslash rm\; i\}\; x$. Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by $\backslash text\{Ce\}\_n(x,\; q)$ and $\backslash text\{Se\}\_n(x,\; q)$, are defined fromMcLachlan (1947), chapter II.

:$$

\begin{align}

\text{Ce}_n(x, q) &= \text{ce}_n({\rm i} x, q). \\

\text{Se}_n(x, q) &= -{\rm i}\,\text{se}_n({\rm i} x, q).

\end{align}

These functions are real-valued when $x$ is real.

A common normalization,Arscott (1964); Iyanaga (1980); Gradshteyn (2007); This is also the normalization used by the computer algebra system Maple. which will be adopted throughout this article, is to demand

:$$

\int_{0}^{2\pi} \text{ce}_n(x, q)^2 dx = \int_{0}^{2\pi} \text{se}_n(x, q)^2 dx = \pi

as well as require $\backslash text\{ce\}\_n(x,\; q)\; \backslash rightarrow\; +\backslash cos\; nx$ and $\backslash text\{se\}\_n(x,\; q)\; \backslash rightarrow\; +\backslash sin\; nx$ as $q\; \backslash rightarrow\; 0$.

{{Main|Floquet theory}}

Many properties of the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called Floquet theory. The central result is Floquet's theorem:

{{math theorem|name=Floquet's theoremArscott (1964), p. 29. | math_statement = Mathieu's equation always has at least one solution $y(x)$ such that $y(x\; +\; \backslash pi)\; =\; \backslash sigma\; y(x)$, where $\backslash sigma$ is a constant which depends on the parameters of the equation and may be real or complex.}}

It is natural to associate the characteristic numbers $a(q)$ with those values of $a$ which result in $\backslash sigma\; =\; \backslash pm\; 1$.It is not true, in general, that a $2\; \backslash pi$ periodic function has the property $y(x\; +\; \backslash pi)\; =\; -y(x)$. However, this turns out to be true for functions which are solutions of Mathieu's equation. Note, however, that the theorem only guarantees the existence of at least one solution satisfying $y(x\; +\; \backslash pi)\; =\; \backslash sigma\; y(x)$, when Mathieu's equation in fact has two independent solutions for any given $a$, $q$. Indeed, it turns out that with $a$ equal to one of the characteristic numbers, Mathieu's equation has only one periodic solution (that is, with period $\backslash pi$ or $2\; \backslash pi$), and this solution is one of the $\backslash text\{ce\}\_n(x,\; q)$, $\backslash text\{se\}\_n(x,\; q)$. The other solution is nonperiodic, denoted $\backslash text\{fe\}\_n(x,\; q)$ and $\backslash text\{ge\}\_n(x,\; q)$, respectively, and referred to as a Mathieu function of the second kind.McLachlan (1951), pp. 141-157, 372 This result can be formally stated as Ince's theorem:

{{math theorem| name = Ince's theoremArscott (1964), p. 34 | math_statement = Define a basically periodic function as one satisfying $y(x\; +\; \backslash pi)\; =\; \backslash pm\; y(x)$. Then, except in the trivial case $q\; =\; 0$, Mathieu's equation never possesses two (independent) basically periodic solutions for the same values of $a$ and $q$.}}

Image:MathieuFloquet.gif

An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form

: $F(a,\; q,\; x)\; =\; \backslash exp(i\; \backslash mu\; \backslash ,x)\; \backslash ,\; P(a,\; q,\; x),$

where $\backslash mu$ is a complex number, the Floquet exponent (or sometimes Mathieu exponent), and $P$ is a complex valued function periodic in $x$ with period $\backslash pi$. An example $P(a,\; q,\; x)$ is plotted to the right.

The Mathieu equation has two parameters. For almost all choices of these parameters, Floquet theory says that any solution either converges to zero or diverges to infinity.

If the Mathieu equation is parameterized as $\backslash ddot\; x\; +\; k(1-m\; \backslash cos(t))x\; =\; 0$, where $k\; \backslash in\; \backslash R,\; m\; \backslash geq\; 0$, then the regions of stability and instability are separated by the following curves:{{sfn|Butikov|2018}}

$$m(k)\; =\; \backslash begin\{cases\}$$

2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, & k<0 ; \\[4pt]

\frac{1}{4} \left[\sqrt{(9-4 k)(13-20 k)}-(9-4 k) \right], & k<\frac{1}{4} ; \\[10pt]

\frac{1}{4} \left[9-4 k \mp \sqrt{(9-4 k)(13-20 k)} \right], & \frac{1}{4}

\sqrt{\frac{2(k-1)(k-4)(k-9)}{k-5}}, & \frac{13}{20}

2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, & k>1 .

\end{cases}

Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if $a$ is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic. The periodic solution is one of the $\backslash text\{ce\}\_n(x,\; q)$ and $\backslash text\{se\}\_n(x,\; q)$, called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either $\backslash text\{fe\}\_n(x,\; q)$ and $\backslash text\{ge\}\_n(x,\; q)$, respectively, and is called a Mathieu function of the second kind (of integral order). The nonperiodic solutions are unstable, that is, they diverge as $z\; \backslash rightarrow\; \backslash pm\; \backslash infty$.McLachlan (1947), p. 144

The second solutions corresponding to the modified Mathieu functions $\backslash text\{Ce\}\_n(x,\; q)$ and $\backslash text\{Se\}\_n(x,\; q)$ are naturally defined as $\backslash text\{Fe\}\_n(x,\; q)\; =\; -i\; \backslash text\{fe\}\_n(xi,\; q)$ and $\backslash text\{Ge\}\_n(x,\; q)\; =\; \backslash text\{ge\}\_n(xi,\; q)$.

Mathieu functions of fractional order can be defined as those solutions $\backslash text\{ce\}\_\{p\}(x,\; q)$ and $\backslash text\{se\}\_\{p\}(x,\; q)$, $p$ a non-integer, which turn into $\backslash cos\; p\; x$ and $\backslash sin\; p\; x$ as $q\; \backslash rightarrow\; 0$. If $p$ is irrational, they are non-periodic; however, they remain bounded as $x\; \backslash rightarrow\; \backslash infty$.

An important property of the solutions $\backslash text\{ce\}\_\{p\}(x,\; q)$ and $\backslash text\{se\}\_\{p\}(x,\; q)$, for $p$ non-integer, is that they exist for the same value of $a$. In contrast, when $p$ is an integer, $\backslash text\{ce\}\_\{p\}(x,\; q)$ and $\backslash text\{se\}\_\{p\}(x,\; q)$ never occur for the same value of $a$. (See Ince's Theorem above.)

These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly.

:

Mathieu functions of the first kind can be represented as Fourier series:

:$$

\begin{align}

\text{ce}_{2n}(x, q) &= \sum_{r=0}^{\infty} A^{(2n)}_{2r}(q) \cos (2 r x) \\

\text{ce}_{2n+1}(x, q) &= \sum_{r=0}^{\infty} A^{(2n+1)}_{2r+1}(q) \cos \left[ (2r+1) x \right] \\

\text{se}_{2n+1}(x, q) &= \sum_{r=0}^{\infty} B^{(2n+1)}_{2r+1}(q) \sin \left[(2r+1) x\right] \\

\text{se}_{2n+2}(x, q) &= \sum_{r=0}^{\infty} B^{(2n+2)}_{2r+2}(q) \sin \left[(2r+2) x\right] \\

\end{align}

The expansion coefficients $A^\{(i)\}\_\{j\}(q)$ and $B^\{(i)\}\_\{j\}(q)$ are functions of $q$ but independent of $x$. By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index. For instance, for each $\backslash text\{ce\}\_\{2n\}$ one findsMcLachlan (1947), p. 28

:$$

\begin{align}

a A_0 - q A_2 &= 0 \\

(a - 4) A_2 - q (A_4 + 2 A_0) &= 0 \\

(a - 4 r^2) A_{2r} - q (A_{2r+2} + A_{2r-2}) &= 0, \quad r \geq 2

\end{align}

Being a second-order recurrence in the index $2r$, one can always find two independent solutions $X\_\{2r\}$ and $Y\_\{2r\}$ such that the general solution can be expressed as a linear combination of the two: $A\_\{2r\}\; =\; c\_1\; X\_\{2r\}\; +\; c\_2\; Y\_\{2r\}$. Moreover, in this particular case, an asymptotic analysisWimp (1984), pp. 83-84 shows that one possible choice of fundamental solutions has the property

:$$

\begin{align}

X_{2r} &= r^{-2r-1} \left(-\frac{e^2 q}{4} \right)^r \left[1 + \mathcal{O}(r^{-1}) \right] \\

Y_{2r} &= r^{2r-1} \left(-\frac{4}{e^2 q} \right)^r \left[1 + \mathcal{O}(r^{-1}) \right]

\end{align}

In particular, $X\_\{2r\}$ is finite whereas $Y\_\{2r\}$ diverges. Writing $A\_\{2r\}\; =\; c\_1\; X\_\{2r\}\; +\; c\_2\; Y\_\{2r\}$, we therefore see that in order for the Fourier series representation of $\backslash text\{ce\}\_\{2n\}$ to converge, $a$ must be chosen such that $c\_2\; =\; 0.$ These choices of $a$ correspond to the characteristic numbers.

In general, however, the solution of a three-term recurrence with variable coefficients

cannot be represented in a simple manner, and hence there is no simple way to determine $a$ from the condition

$c\_2\; =\; 0$. Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients $A\_\{2r\}$ by numerically iterating the recurrence towards increasing $r$. The reason is that as long as $a$ only approximates a characteristic number, $c\_2$ is not identically $0$ and the divergent solution $Y\_\{2r\}$ eventually dominates for large enough $r$.

To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a continued fraction expansion,McLachlan (1947) casting the recurrence as a matrix eigenvalue problem,Chaos-Cador and Ley-Koo (2001) or implementing a backwards recurrence algorithm. The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions.Temme (2015), p. 234

In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as Mathematica, Maple, MATLAB, and SciPy. For small values of $q$ and low order $n$, they can also be expressed perturbatively as power series of $q$, which can be useful in physical applications.Müller-Kirsten (2012), pp. 420-428

There are several ways to represent Mathieu functions of the second kind.Meixner and Schäfke (1954); McLachlan (1947) One representation is in terms of Bessel functions:Malits (2010)

:$$

\begin{align}

\text{fe}_{2n}(x, q) &= -\frac{\pi \gamma_n}{2} \sum_{r=0}^{\infty} (-1)^{r+n} A^{(2n)}_{2r}(-q) \ \text{Im}[J_r(\sqrt{q}e^{ix}) Y_r(\sqrt{q}e^{-ix})], \quad \text{where } \gamma_n = \left\{

\begin{array}{cc}

\sqrt{2}, & \text{ if } n = 0 \\

2n, & \text{ if } n \geq 1

\end{array}

\right. \\

\text{fe}_{2n+1}(x, q) &= \frac{\pi \sqrt{q}}{2} \sum_{r=0}^{\infty} (-1)^{r+n} A^{(2n+1)}_{2r+1}(-q) \ \text{Im}[J_r(\sqrt{q}e^{ix}) Y_{r+1}(\sqrt{q}e^{-ix}) + J_{r+1}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \\

\text{ge}_{2n+1}(x, q) &= -\frac{\pi \sqrt{q}}{2} \sum_{r=0}^{\infty} (-1)^{r+n} B^{(2n+1)}_{2r+1}(-q) \ \text{Re}[J_r(\sqrt{q}e^{ix}) Y_{r+1}(\sqrt{q}e^{-ix}) - J_{r+1}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})] \\

\text{ge}_{2n+2}(x, q) &= -\frac{\pi q}{4(n+1)} \sum_{r=0}^{\infty} (-1)^{r+n} B^{(2n+2)}_{2r+2}(-q) \ \text{Re}[J_r(\sqrt{q}e^{ix}) Y_{r+2}(\sqrt{q}e^{-ix}) - J_{r+2}(\sqrt{q}e^{ix}) Y_r(\sqrt{q} e^{-ix})]

\end{align}

where $n,\; q\; >\; 0$, and $J\_r(x)$ and $Y\_r(x)$ are Bessel functions of the first and second kind.

A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series.Jin and Zhang (1996) For large $n$ and $q$, the form of the series must be chosen carefully to avoid subtraction errors.Van Buren and Boisvert (2007)Bibby and Peterson (2013)

There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other special functions, the solutions of Mathieu's equation cannot in general be expressed in terms of hypergeometric functions. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable $t=\backslash cos(x)$:

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

Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type.

File:Mathieu functions sample.png

File:Ce1 vary q.png

For small $q$, $\backslash text\{ce\}\_\{n\}$ and $\backslash text\{se\}\_\{n\}$ behave similarly to $\backslash cos\; nx$ and $\backslash sin\; nx$. For arbitrary $q$, they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real $q$, $\backslash text\{ce\}\_\{m\}(x,\; q)$ and $\backslash text\{se\}\_\{m+1\}(x,\; q)$ have exactly $m$ simple zeros in , and as $q\; \backslash rightarrow\; \backslash infty$ the zeros cluster about $x\; =\; \backslash pi\; /\; 2$.Meixner and Schäfke (1954), p.134McLachlan (1947), pp. 234–235

For $q\; >\; 0$ and as $x\; \backslash rightarrow\; \backslash infty$ the modified Mathieu functions tend to behave as damped periodic functions.

In the following, the $A$ and $B$ factors from the Fourier expansions for $\backslash text\{ce\}\_\{n\}$ and $\backslash text\{se\}\_\{n\}$ may be referenced (see Explicit representation and computation). They depend on $q$ and $n$ but are independent of $x$.

Due to their parity and periodicity, $\backslash text\{ce\}\_n$ and $\backslash text\{se\}\_n$ have simple properties under reflections and translations by multiples of $\backslash pi$:

:$$

\begin{align}

&\text{ce}_n(x + \pi) = (-1)^n \text{ce}_n(x) \\

&\text{se}_n(x + \pi) = (-1)^n \text{se}_n(x) \\

&\text{ce}_n(x + \pi / 2) = (-1)^n \text{ce}_n(-x + \pi / 2) \\

&\text{se}_{n+1}(x + \pi / 2) = (-1)^n \text{se}_{n+1}(-x + \pi / 2)

\end{align}

One can also write functions with negative $q$ in terms of those with positive $q$:Gradshteyn (2007), p. 953

:$$

\begin{align}

&\text{ce}_{2n+1}(x, -q) = (-1)^n \text{se}_{2n+1}(-x + \pi/2, q) \\

&\text{ce}_{2n+2}(x, -q) = (-1)^n \text{ce}_{2n+2}(-x + \pi/2, q) \\

&\text{se}_{2n+1}(x, -q) = (-1)^n \text{ce}_{2n+1}(-x + \pi/2, q) \\

&\text{se}_{2n+2}(x, -q) = (-1)^n \text{se}_{2n+2}(-x + \pi/2, q)

\end{align}

Moreover,

:$$

\begin{align}

&a_{2n+1}(q) = b_{2n+1}(-q)\\

&b_{2n+2}(q) = b_{2n+2}(-q)

\end{align}

Like their trigonometric counterparts $\backslash cos\; nx$ and $\backslash sin\; nx$, the periodic Mathieu functions $\backslash text\{ce\}\_n(x,\; q)$ and $\backslash text\{se\}\_n(x,\; q)$ satisfy orthogonality relations

:$$

\begin{align}

&\int_0^{2\pi} \text{ce}_n \text{ce}_m \,dx = \int_0^{2\pi} \text{se}_n \text{se}_m \, dx = \delta_{nm} \pi \\

&\int_0^{2\pi} \text{ce}_n \text{se}_m \,dx = 0

\end{align}

Moreover, with $q$ fixed and $a$ treated as the eigenvalue, the Mathieu equation is of Sturm–Liouville form. This implies that the eigenfunctions $\backslash text\{ce\}\_n(x,\; q)$ and $\backslash text\{se\}\_n(x,\; q)$ form a complete set, i.e. any $\backslash pi$- or $2\; \backslash pi$-periodic function of $x$ can be expanded as a series in $\backslash text\{ce\}\_n(x,\; q)$ and $\backslash text\{se\}\_n(x,\; q)$.

Solutions of Mathieu's equation satisfy a class of integral identities with respect to kernels $\backslash chi(x,\; x\text{'})$ that are solutions of

:$$

\frac{\partial^2 \chi}{\partial x^2} - \frac{\partial^2 \chi}{\partial x'^2} = 2 q \left(\cos 2x - \cos 2x' \right) \chi

More precisely, if $\backslash phi(x)$ solves Mathieu's equation with given $a$ and $q$, then the integral

:$$

\psi(x) \equiv \int_C \chi(x, x') \phi(x') dx'

where $C$ is a path in the complex plane, also solves Mathieu's equation with the same $a$ and $q$, provided the following conditions are met:Arscott (1964), pp. 40-41

• $\backslash chi(x,\; x\text{'})$ solves $\backslash frac\{\backslash partial^2\; \backslash chi\}\{\backslash partial\; x^2\}\; -\; \backslash frac\{\backslash partial^2\; \backslash chi\}\{\backslash partial\; x\text{'}^2\}\; =\; 2\; q\; \backslash left(\backslash cos\; 2x\; -\; \backslash cos\; 2x\text{'}\; \backslash right)\; \backslash chi$
• In the regions under consideration, $\backslash psi(x)$ exists and $\backslash chi(x,\; x\text{'})$ is analytic
• $\backslash left(\; \backslash phi\; \backslash frac\{\backslash partial\; \backslash chi\}\{\backslash partial\; x\text{'}\}\; -\; \backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\text{'}\}\; \backslash chi\; \backslash right)$ has the same value at the endpoints of $C$

Using an appropriate change of variables, the equation for $\backslash chi$ can be transformed into the wave equation and solved. For instance, one solution is $\backslash chi(x,\; x\text{'})\; =\; \backslash sinh\; (2\; q^\{1/2\}\; \backslash sin\; x\; \backslash sin\; x\text{'}\; )$. Examples of identities obtained in this way areGradshteyn (2007), pp. 763–765

:$$

\begin{align}

\text{se}_{2n+1}(x, q) &= \frac{\text{se}'_{2n+1}(0, q)}{\pi q^{1/2} B_1^{(2n+1)}} \int_0^{\pi} \sinh (2 q^{1/2} \sin x \sin x' ) \text{se}_{2n+1}(x', q) dx' \qquad (q > 0) \\

\text{Ce}_{2n}(x, q) &= \frac{\text{ce}_{2n}(\pi / 2, q)}{\pi A_0^{(2n)}} \int_0^{\pi} \cos (2 q^{1/2} \cosh x \cos x' ) \text{ce}_{2n}(x', q) dx' \qquad \ \ \ (q > 0)

\end{align}

Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions.Arscott (1964), p. 86

There also exist integral relations between functions of the first and second kind, for instance:

:$$

\text{fe}_{2n}(x, q) = 2n \int_0^x \text{ce}_{2n}(\tau, -q) \ J_0 \left( \sqrt{2q(\cos 2x - \cos 2\tau)} \right) d \tau, \qquad n \geq 1

valid for any complex $x$ and real $q$.

The following asymptotic expansions hold for $q\; >\; 0$, $\backslash text\{Im\}(x)\; =\; 0$, $\backslash text\{Re\}(x)\; \backslash rightarrow\; \backslash infty$, and $2\; q^\{1/2\}\; \backslash cosh\; x\; \backslash simeq\; q^\{1/2\}\; e^x$:McLachlan (1947), chapter XI

:$$

\begin{align}

\text{Ce}_{2n}(x,q) &\sim \left(\frac{2}{\pi q^{1/2}} \right)^{1/2} \frac{\text{ce}_{2n}(0, q)\text{ce}_{2n}(\pi/2, q)}{A_0^{(2n)}} \cdot e^{-x/2} \sin \left( q^{1/2} e^x + \frac{\pi}{4} \right) \\

\text{Ce}_{2n+1}(x,q) &\sim \left(\frac{2}{\pi q^{3/2}} \right)^{1/2} \frac{\text{ce}_{2n+1}(0, q)\text{ce}'_{2n+1}(\pi/2, q)}{A_1^{(2n+1)}} \cdot e^{-x/2} \cos \left( q^{1/2} e^x + \frac{\pi}{4} \right) \\

\text{Se}_{2n+1}(x,q) &\sim -\left(\frac{2}{\pi q^{3/2}} \right)^{1/2} \frac{\text{se}'_{2n+1}(0, q)\text{se}_{2n+1}(\pi/2, q)}{B_1^{(2n+1)}} \cdot e^{-x/2} \cos \left( q^{1/2} e^x + \frac{\pi}{4} \right) \\

\text{Se}_{2n+2}(x,q) &\sim \left(\frac{2}{\pi q^{5/2}} \right)^{1/2} \frac{\text{se}'_{2n+2}(0, q)\text{se}'_{2n+2}(\pi/2, q)}{B_2^{(2n+2)}} \cdot e^{-x/2} \sin \left( q^{1/2} e^x + \frac{\pi}{4} \right)

\end{align}

Thus, the modified Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for $\backslash text\{Fe\}\_\{n\}$ and $\backslash text\{Ge\}\_\{n\}$; these also decay exponentially for large real argument.

For the even and odd periodic Mathieu functions $ce,\; se$ and the associated characteristic numbers $a$ one can also derive asymptotic expansions for large $q$.McLachlan (1947), p. 237; Dingle and Müller (1962); Müller (1962); Dingle and Müller(1964) For the characteristic numbers in particular, one has with $N$ approximately an odd integer, i.e. $N\backslash approx\; N\_0=\; 2n+1,\; n\; =1,2,3,...,$

:$$

\begin{align}

a(N) ={}& -2q + 2q^{1/2}N -\frac{1}{2^3}(N^2+1) -\frac{1}{2^7q^{1/2}}N(N^2 +3) -\frac{1}{2^{12}q}(5N^4 + 34N^2 +9) \\

& -\frac{1}{2^{17}q^{3/2}}N(33N^4 +410N^2 +405) -\frac{1}{2^{20}q^2}(63N^6 + 1260N^4 + 2943N^2 +41807) +

\mathcal{O}(q^{-5/2})

\end{align}

Observe the symmetry here in replacing $q^\{1/2\}$ and $N$ by $-q^\{1/2\}$ and $-\; N$, which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order $|q|^\{-7/2\}$.Dingle and Müller (1962) Here $N$ is only approximately an odd integer because in the limit of $q\backslash to\; \backslash infty$ all minimum segments of the periodic potential $\backslash cos\; 2x$ become effectively independent harmonic oscillators (hence $N\_0$ an odd integer). By decreasing $q$, tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers $a\; \backslash to\; a\_\{\backslash mp\}$ (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions (in quantum mechanics this provides the splitting of the eigenvalues into energy bands).Müller-Kirsten (2012) The boundary conditions are:

:$\backslash left(\backslash frac\{dce\_\{N\_0-1\}\}\{dx\}\backslash right)\_\{\backslash pi/2\}=0,\backslash ;\backslash ;\; ce\_\{N\_0\}(\backslash pi/2)=0,\; \backslash ;\backslash ;\; \backslash left(\backslash frac\{dse\_\{N\_0\}\}\{dx\}\backslash right)\_\{\backslash pi/2\}=0,\; \backslash ;\backslash ;\; se\_\{N\_0+1\}(\backslash pi/2)=0.$

Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for $a$ one obtains

:$N-N\_0\; =\; \backslash mp\; 2\backslash left(\backslash frac\{2\}\{\backslash pi\}\backslash right)^\{1/2\}\; \backslash frac\{(16q^\{1/2\})^\{N\_0/2\}e^\{-4q^\{1/2\}\}\}\{[\backslash frac\{1\}\{2\}(N\_0-1)]!\}\backslash left[1-\backslash frac\{3(N\_0^2+1)\}\{2^6q^\{1/2\}\}\; +\; \backslash frac\{1\}\{2^\{13\}q\}(9N\_0^4\; -40N\_0^3\; +18N\_0^2\; -136N\_0\; +\; 9)\; +\; \backslash dots\; \backslash right].$

The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.

:$a(N)\; =\; a(N\_0)\; +\; (N-N\_0)\; \backslash left(\backslash frac\{\backslash partial\; a\}\{\backslash partial\; N\}\backslash right)\_\{N\_0\}\; +\; \backslash cdots\; .$

Insertion of the appropriate expressions above yields the result

:$\backslash begin\{align\}$

a(N)\to a_{\mp}(N_0) = {} & -2q + 2q^{1/2}N_0 -\frac{1}{2^3}(N^2_0+1) - \frac{1}{2^7q^{1/2}}N_0(N_0^2+3) -

\frac{1}{2^{12}q}(5N_0^4+34N_0^2 +9) - \cdots \\

& \mp \frac{(16q^{1/2})^{N_0/2+1}e^{-4q^{1/2}}}{(8\pi)^{1/2}[\frac{1}{2}(N_0-1)]!}\bigg[1-\frac{N_0}{2^6q^{1/2}}(3N_0^2+8N_0+3) + \cdots\bigg].

\end{align}

For $N\_0=1,3,5,\backslash dots$ these are the eigenvalues associated with the even Mathieu eigenfunctions $ce\_\{N\_0\}$ or $ce\_\{N\_0-1\}$ (i.e. with upper, minus sign) and odd Mathieu eigenfunctions $se\_\{N\_0+1\}$ or

$se\_\{N\_0\}$ (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in or.

Similar asymptotic expansions can be obtained for the solutions of other periodic differential equations, as for Lamé functions and prolate and oblate spheroidal wave functions.

Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed below.

Mathieu functions arise when separation of variables in elliptic coordinates is applied to 1) the Laplace equation in 3 dimensions, and 2) the Helmholtz equation in either 2 or 3 dimensions. Since the Helmholtz equation is a prototypical equation for modeling the spatial variation of classical waves, Mathieu functions can be used to describe a variety of wave phenomena. For instance, in computational electromagnetics they can be used to analyze the scattering of electromagnetic waves off elliptic cylinders, and wave propagation in elliptic waveguides.Bibby and Peterson (2013); Barakat (1963); Sebak and Shafai (1991); Kretzschmar (1970) In general relativity, an exact plane wave solution to the Einstein field equation can be given in terms of Mathieu functions.

More recently, Mathieu functions have been used to solve a special case of the Smoluchowski equation, describing the steady-state statistics of self-propelled particles.Solon et al (2015)

The remainder of this section details the analysis for the two-dimensional Helmholtz equation.see Willatzen and Voon (2011), pp. 61–65 In rectangular coordinates, the Helmholtz equation is

:$$

\left(\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} \right) \psi + k^2 \psi = 0,

Elliptic coordinates are defined by

:$$

\begin{align}

x &= c \cosh \mu \cos \nu \\

y &= c \sinh \mu \sin \nu

\end{align}

where , , and $c$ is a positive constant. The Helmholtz equation in these coordinates is

:$$

\frac{1}{c^2 (\sinh^2 \mu + \sin^2 \nu)} \left(\frac{\partial^2}{\partial \mu^2} +\frac{\partial^2}{\partial \nu^2} \right) \psi + k^2 \psi = 0

The constant $\backslash mu$ curves are confocal ellipses with focal length $c$; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via $\backslash psi(\backslash mu,\; \backslash nu)\; =\; F(\backslash mu)\; G(\backslash nu)$ yields the Mathieu equations

:$$

\begin{align}

&\frac{d^2F}{d \mu^2} - \left(a - \frac{c^2 k^2}{2} \cosh 2 \mu \right) F = 0 \\

&\frac{d^2G}{d \nu^2} + \left(a - \frac{c^2 k^2}{2} \cos 2\nu \right) G = 0 \\

\end{align}

where $a$ is a separation constant.

As a specific physical example, the Helmholtz equation can be interpreted as describing normal modes of an elastic membrane under uniform tension. In this case, the following physical conditions are imposed:McLachlan (1947), pp. 294–297

• Periodicity with respect to $\backslash nu$, i.e. $\backslash psi(\backslash mu,\; \backslash nu)\; =\; \backslash psi(\backslash mu,\; \backslash nu\; +\; 2\; \backslash pi)$
• Continuity of displacement across the interfocal line: $\backslash psi(0,\; \backslash nu)\; =\; \backslash psi(0,\; -\backslash nu)$
• Continuity of derivative across the interfocal line: $\backslash psi\_\{\backslash mu\}(0,\; \backslash nu)\; =\; -\backslash psi\_\{\backslash mu\}(0,\; -\backslash nu)$

For given $k$, this restricts the solutions to those of the form $\backslash text\{Ce\}\_\{n\}(\backslash mu,\; q)\backslash text\{ce\}\_n(\backslash nu,\; q)$ and $\backslash text\{Se\}\_\{n\}(\backslash mu,\; q)\backslash text\{se\}\_n(\backslash nu,\; q)$, where $q\; =\; c^2\; k^2\; /\; 4$. This is the same as restricting allowable values of $a$, for given $k$. Restrictions on $k$ then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by $\backslash mu\; =\; \backslash mu\_0\; >\; 0$. For instance, clamping the membrane at $\backslash mu\; =\; \backslash mu\_0$ imposes $\backslash psi(\backslash mu\_0,\; \backslash nu)\; =\; 0$, which in turn requires

:$$

\begin{align}

\text{Ce}_{n}(\mu_0, q) = 0 \\

\text{Se}_{n}(\mu_0, q) = 0

\end{align}

These conditions define the normal modes of the system.

In dynamical problems with periodically varying forces, the equation of motion sometimes takes the form of Mathieu's equation. In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics.Meixner and Schäfke (1954), pp. 324–343 A classic example along these lines is the inverted pendulum.Ruby (1996) Other examples are

• vibrations of a string with periodically varying tension
• stability of railroad rails as trains drive over them
• seasonally forced population dynamics
• the phenomenon of parametric resonance in forced oscillators
• motion of ions in a quadrupole ion trapMarch (1997)
• the Stark effect for a rotating electric dipole
• the Floquet theory of the stability of limit cycles
• analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with periodic noise term{{Cite book |last1=Barna |first1=Imre Ferenc |title=Differential and Difference Equations with Applications, ICDDEA 2019, Lisbon, Portugal, July 1–5 Conference proceedings |last2=Bognár |first2=G. |last3=Mátyás |first3=L. |last4=Guedda |first4=M. |last5=Hriczó |first5=K. |date=2020 |publisher=Springer |isbn=9783030563226 |editor-last=Pinelas |editor-first=S. |pages=239–254 |chapter=Analytic Traveling-Wave Solutions of the Kardar-Parisi-Zhang Interface Growing Equation with Different Kind of Noise Terms |arxiv=1908.09615 |editor-last2=Graef |editor-first2=John R. |editor-last3=Hilger |editor-first3=S. |editor-last4=Kloeden |editor-first4=P |editor-last5=Shinas |editor-first5=C. }}

Mathieu functions play a role in certain quantum mechanical systems, particularly those with spatially periodic potentials such as the quantum pendulum and crystalline lattices.

The modified Mathieu equation also arises when describing the quantum mechanics of singular potentials. For the particular singular potential $V(r)\; =\; g^2/r^4$ the radial Schrödinger equation

: $\backslash frac\{d^2y\}\{dr^2\}\; +\; \backslash left[\; k^2\; -\; \backslash frac\{\backslash ell(\backslash ell+1)\}\{r^2\}\; -\backslash frac\{g^2\}\{r^4\}\; \backslash right]\; y\; =\; 0$

can be converted into the equation

: $\backslash frac\{d^2\backslash varphi\}\{dz^2\}\; +\; \backslash left[\; 2h^2\; \backslash cosh\; 2z\; -\; \backslash left(\backslash ell+\backslash frac\; 1\; 2\; \backslash right)^2\backslash right]\; \backslash varphi\; =\; 0.$

The transformation is achieved with the following substitutions

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

By solving the Schrödinger equation (for this particular potential) in terms of solutions of the modified Mathieu equation, scattering properties such as the S-matrix and the absorptivity can be obtained.Müller-Kirsten (2006)

Originally the Schrödinger equation with cosine function was solved in 1928 by Strutt.{{Cite journal |last=Strutt |first=M.J.O. |date=1928 |title=Zur Wellenmechanik des Atomgitters |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19283911006 |journal= Annalen der Physik|volume=86 |issue=10 |pages=319–324 |doi=10.1002/andp.19283911006|bibcode=1928AnP...391..319S |url-access=subscription }}

• Almost Mathieu operator
• Bessel function
• Hill differential equation
• Inverted pendulum
• Lamé function
• List of mathematical functions
• Monochromatic electromagnetic plane wave

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{{Refend}}

• {{MathWorld|urlname=MathieuFunction |title=Mathieu function}}
• [http://functions.wolfram.com/MathieuandSpheroidalFunctions/ List of equations and identities for Mathieu Functions] functions.wolfram.com
• {{springer|title=Mathieu functions|id=p/m062760}}
• Timothy Jones, [http://www.physics.drexel.edu/~tim/open/mat/mat.html Mathieu's Equations and the Ideal rf-Paul Trap] (2006)
• [http://eqworld.ipmnet.ru/en/solutions/ode/ode0234.pdf Mathieu equation], [http://eqworld.ipmnet.ru/en/ EqWorld]
• [http://dlmf.nist.gov/28 NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation]

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Category:Ordinary differential equations

Category:Special functions