Mathieu function

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac{d^2y}{dx^2} + (a - 2q\cos(2x))y = 0, where are real-valued parameters. Since we may add to to change the sign of , it is a usual convention to set . They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads. They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation (PDE) boundary value problems possessing elliptic symmetry.

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation \frac{d^2y}{dx^2} + (a - 2q\cos(2x))y = 0, where are real-valued parameters. Since we may add to to change the sign of , it is a usual convention to set . They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads. They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation (PDE) boundary value problems possessing elliptic symmetry. 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 \pi - or 2\pi -periodic solutions, which exist only for special values of a and q . 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, \ldots . The corresponding functions are denoted \text{ce}_n(x, q) and \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. \text{ce}_n(x, q) and \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 \text{ce}_n(x, q) and \text{se}_n(x, q) become proportional to \cos nx and \sin nx as q \rightarrow 0 . With n being an integer, this gives rise to the classification of \text{ce}_n and \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. \frac{d^2y}{dx^2} - (a - 2q\cosh 2x)y = 0, which can be related to the original Mathieu equation by taking x \to \pm {\rm i} x . Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by \text{Ce}_n(x, q) and \text{Se}_n(x, q) , are defined from \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. \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 \text{ce}_n(x, q) \rightarrow +\cos nx and \text{se}_n(x, q) \rightarrow +\sin nx as q \rightarrow 0 . It is natural to associate the characteristic numbers a(q) with those values of a which result in \sigma = \pm 1 . Note, however, that the theorem only guarantees the existence of at least one solution satisfying y(x + \pi) = \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 \pi or 2 \pi ), and this solution is one of the \text{ce}_n(x, q) , \text{se}_n(x, q) . The other solution is nonperiodic, denoted \text{fe}_n(x, q) and \text{ge}_n(x, q) , respectively, and referred to as a Mathieu function of the second kind. This result can be formally stated as Ince's theorem: An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form F(a, q, x) = \exp(i \mu \,x) \, P(a, q, x), where \mu is a complex number, the Floquet exponent (or sometimes Mathieu exponent), and P is a complex valued function periodic in x with period \pi . An example P(a, q, x) is plotted to the right. If the Mathieu equation is parameterized as \ddot x + k(1-m \cos(t))x = 0 , where k \in \R, m \geq 0 , then the regions of stability and instability are separated by the following curves: m(k) = \begin{cases} 2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, & k 1 . \end{cases} The second solutions corresponding to the modified Mathieu functions \text{Ce}_n(x, q) and \text{Se}_n(x, q) are naturally defined as \text{Fe}_n(x, q) = -i \text{fe}_n(xi, q) and \text{Ge}_n(x, q) = \text{ge}_n(xi, q) . \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 \text{ce}_{2n} one finds \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 analysis 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 \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, casting the recurrence as a matrix eigenvalue problem, 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. 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. \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. 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=\cos(x) : (1-t^2)\frac{d^2y}{dt^2} - t\, \frac{d y}{dt} + (a + 2q (1- 2t^2)) \, y=0. Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type. For small q , \text{ce}_{n} and \text{se}_{n} behave similarly to \cos nx and \sin nx . For arbitrary q , they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real q , \text{ce}_{m}(x, q) and \text{se}_{m+1}(x, q) have exactly m simple zeros in 0 , and as q \rightarrow \infty the zeros cluster about x = \pi / 2 . For q > 0 and as x \rightarrow \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 \text{ce}_{n} and \text{se}_{n} may be referenced (see Explicit representation and computation). They depend on q and n but are independent of x . &\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 : \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} &\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 \text{ce}_n(x, q) and \text{se}_n(x, q) form a complete set, i.e. any \pi - or 2 \pi -periodic function of x can be expanded as a series in \text{ce}_n(x, q) and \text{se}_n(x, q) . \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 \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: \chi(x, x') solves \frac{\partial^2 \chi}{\partial x^2} - \frac{\partial^2 \chi}{\partial x'^2} = 2 q \left(\cos 2x - \cos 2x' \right) \chi In the regions under consideration, \psi(x) exists and \chi(x, x') is analytic \left( \phi \frac{\partial \chi}{\partial x'} - \frac{\partial \phi}{\partial x'} \chi \right) has the same value at the endpoints of C Using an appropriate change of variables, the equation for \chi can be transformed into the wave equation and solved. For instance, one solution is \chi(x, x') = \sinh (2 q^{1/2} \sin x \sin x' ) . Examples of identities obtained in this way are \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. 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 , \text{Im}(x) = 0 , \text{Re}(x) \rightarrow \infty , and 2 q^{1/2} \cosh x \simeq q^{1/2} e^x : \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 \text{Fe}_{n} and \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 . For the characteristic numbers in particular, one has with N approximately an odd integer, i.e. N\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} . Here N is only approximately an odd integer because in the limit of q\to \infty all minimum segments of the periodic potential \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 \to a_{\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). The boundary conditions are: \left(\frac{dce_{N_0-1}}{dx}\right)_{\pi/2}=0,\;\; ce_{N_0}(\pi/2)=0, \;\; \left(\frac{dse_{N_0}}{dx}\right)_{\pi/2}=0, \;\; se_{N_0+1}(\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 = \mp 2\left(\frac{2}{\pi}\right)^{1/2} \frac{(16q^{1/2})^{N_0/2}e^{-4q^{1/2}}}{[\frac{1}{2}(N_0-1)]!}\left[1-\frac{3(N_0^2+1)}{2^6q^{1/2}} + \frac{1}{2^{13}q}(9N_0^4 -40N_0^3 +18N_0^2 -136N_0 + 9) + \dots \right]. The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e. a(N) = a(N_0) + (N-N_0) \left(\frac{\partial a}{\partial N}\right)_{N_0} + \cdots . Insertion of the appropriate expressions above yields the result \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,\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. The asymptotic nature of the asymptotic solutions of the Mathieu equation has been investigated in detail by investigating the large order behaviour of the expansions. \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 0 \leq \mu , 0 \leq \nu , 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 \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 \psi(\mu, \nu) = F(\mu) G(\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: Periodicity with respect to \nu , i.e. \psi(\mu, \nu) = \psi(\mu, \nu + 2 \pi) Continuity of displacement across the interfocal line: \psi(0, \nu) = \psi(0, -\nu) Continuity of derivative across the interfocal line: \psi_{\mu}(0, \nu) = -\psi_{\mu}(0, -\nu) For given k , this restricts the solutions to those of the form \text{Ce}_{n}(\mu, q)\text{ce}_n(\nu, q) and \text{Se}_{n}(\mu, q)\text{se}_n(\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 \mu = \mu_0 > 0 . For instance, clamping the membrane at \mu = \mu_0 imposes \psi(\mu_0, \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. 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 \frac{d^2y}{dr^2} + \left[ k^2 - \frac{\ell(\ell+1)}{r^2} -\frac{g^2}{r^4} \right] y = 0 can be converted into the equation \frac{d^2\varphi}{dz^2} + \left[ 2h^2 \cosh 2z - \left(\ell+\frac 1 2 \right)^2\right] \varphi = 0. The transformation is achieved with the following substitutions y = r^{1/2} \varphi, r=\gamma e^z, \gamma = \frac{ig}{h}, h^2 = ikg, h = e^{I\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. Originally the Schrödinger equation with cosine function was solved in 1928 by Strutt. ==See also== Almost Mathieu operator Bessel function Hill differential equation Inverted pendulum Lamé function List of mathematical functions Monochromatic electromagnetic plane wave ==Notes== ==References== Note: Reprinted lithographically in Great Britain at the University Press, Oxford, 1951 from corrected sheets of the (1947) first edition. (free online access to the appendix on Mathieu functions) Ho-Chul Shin, "Numerical Review of Mathieu Function Programs for Integer Orders and Real Parameters", SIAM Review, Vol.67, No.4 (2025) ==External links== List of equations and identities for Mathieu Functions functions.wolfram.com Timothy Jones, Mathieu's Equations and the Ideal rf-Paul Trap (2006) Mathieu equation, EqWorld NIST Digital Library of Mathematical Functions: Mathieu Functions and Hill's Equation

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