>+Classification of Mathieu functions[McLachlan (1947), p. 372]
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Integral | | |
Integral | | |
Fractional
( non-integral) | | |
Explicit representation and computation
First kind
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 \\
\text{se}_{2n+1}(x, q) &= \sum_{r=0}^{\infty} B^{(2n+1)}_{2r+1}(q) \sin \left(2r+1) \\
\text{se}_{2n+2}(x, q) &= \sum_{r=0}^{\infty} B^{(2n+2)}_{2r+2}(q) \sin \left(2r+2) \\
\end{align}
The expansion coefficients and are functions of but independent of . By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index. For instance, for each one finds[McLachlan (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 , one can always find two independent solutions and such that the general solution can be expressed as a linear combination of the two: . Moreover, in this particular case, an asymptotic analysis[Wimp (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 \left1 \\
Y_{2r} &= r^{2r-1} \left(-\frac{4}{e^2 q} \right)^r \left1
\end{align}
In particular, is finite whereas diverges. Writing , we therefore see that in order for the Fourier series representation of to converge, must be chosen such that These choices of 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 from the condition
. Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients by numerically iterating the recurrence towards increasing . The reason is that as long as only approximates a characteristic number, is not identically and the divergent solution eventually dominates for large enough .
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 and low order , they can also be expressed perturbatively as power series of , which can be useful in physical applications.[Müller-Kirsten (2012), pp. 420-428]
Second kind
There are several ways to represent Mathieu functions of the second kind.[Meixner and Schäfke (1954); McLachlan (1947)] One representation is in terms of :[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}), \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}) \\
\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}) \\
\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})
\end{align}
where , and and are Bessel functions of the first and second kind.
Modified functions
A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel function product series.[Jin and Zhang (1996)] For large and , the form of the series must be chosen carefully to avoid subtraction errors.[Van Buren and Boisvert (2007)][Bibby and Peterson (2013)]
Properties
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 :
Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type.
Qualitative behavior
For small , and behave similarly to and . For arbitrary , they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real , and have exactly in , and as the zeros cluster about .[Meixner and Schäfke (1954), p.134][McLachlan (1947), pp. 234–235]
For and as the modified Mathieu functions tend to behave as damped periodic functions.
In the following, the and factors from the Fourier expansions for and may be referenced (see Explicit representation and computation). They depend on and but are independent of .
Reflections and translations
Due to their parity and periodicity, and have simple properties under reflections and translations by multiples of :
\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 in terms of those with positive :[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}
Orthogonality and completeness
Like their trigonometric counterparts and , the periodic Mathieu functions and 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 fixed and treated as the eigenvalue, the Mathieu equation is of Sturm–Liouville form. This implies that the eigenfunctions and form a complete set, i.e. any - or -periodic function of can be expanded as a series in and .
Integral identities
Solutions of Mathieu's equation satisfy a class of integral identities with respect to Integral kernel 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 solves Mathieu's equation with given and , then the integral
\psi(x) \equiv \int_C \chi(x, x') \phi(x') dx'
where is a path in the complex plane, also solves Mathieu's equation with the same and , provided the following conditions are met:[Arscott (1964), pp. 40-41]
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solves
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In the regions under consideration, exists and is analytic
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has the same value at the endpoints of
Using an appropriate change of variables, the equation for can be transformed into the wave equation and solved. For instance, one solution is . Examples of identities obtained in this way are[Gradshteyn (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 and real .
Asymptotic expansions
The following asymptotic expansions hold for , , , and :[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 and ; these also decay exponentially for large real argument.
For the even and odd periodic Mathieu functions and the associated characteristic numbers one can also derive asymptotic expansions for large .[McLachlan (1947), p. 237; Dingle and Müller (1962); Müller (1962); Dingle and Müller(1964)] For the characteristic numbers in particular, one has with approximately an odd integer, i.e.
\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 and by and , which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order .[Dingle and Müller (1962)] Here is only approximately an odd integer because in the limit of all minimum segments of the periodic potential become effectively independent harmonic oscillators (hence an odd integer). By decreasing , tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers (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:
Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for one obtains
The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.
Insertion of the appropriate expressions above yields the result
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)!}\bigg1-\frac{N_0}{2^6q^{1/2}}(3N_0^2+8N_0+3).
\end{align}
For these are the eigenvalues associated with the even Mathieu eigenfunctions or (i.e. with upper, minus sign) and odd Mathieu eigenfunctions or
(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.
Applications
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.
Partial differential equations
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 .[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 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 curves are confocal ellipses with focal length ; hence, these coordinates are convenient for solving the Helmholtz equation on domains with elliptic boundaries. Separation of variables via 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 is a separation constant.
As a specific physical example, the Helmholtz equation can be interpreted as describing of an elastic membrane under uniform tension. In this case, the following physical conditions are imposed:[McLachlan (1947), pp. 294–297]
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Periodicity with respect to , i.e.
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Continuity of displacement across the interfocal line:
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Continuity of derivative across the interfocal line:
For given , this restricts the solutions to those of the form and , where . This is the same as restricting allowable values of , for given . Restrictions on then arise due to imposition of physical conditions on some bounding surface, such as an elliptic boundary defined by . For instance, clamping the membrane at imposes , 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.
Dynamical problems
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
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vibrations of a string with periodically varying tension
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stability of railroad rails as trains drive over them
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seasonally forced population dynamics
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the phenomenon of parametric resonance in forced
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motion of ions in a quadrupole ion trap
[March (1997)]
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the Stark effect for a rotating electric dipole
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the Floquet theory of the stability of limit cycles
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analytic Traveling wave solutions of the Kardar-Parisi-Zhang interface growing equation with periodic noise term
Quantum mechanics
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 the radial Schrödinger equation
can be converted into the equation
The transformation is achieved with the following substitutions
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 Absorptance can be obtained.[Müller-Kirsten (2006)]
Originally the Schrödinger equation with cosine function was solved in 1928 by Strutt.
See also
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Almost Mathieu operator
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Bessel function
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Hill differential equation
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Inverted pendulum
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Lamé function
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List of mathematical functions
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Monochromatic electromagnetic plane wave
Notes
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Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition.
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(free online access to the appendix on Mathieu functions)
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External links