28 Mathieu Functions and Hill’s Equation Applications 28.31 Equations of Whittaker–Hill and Ince 28.33 Physical Applications

§28.32 Mathematical Applications

Keywords:: Mathieu functions
Referenced by:: ¶ ‣ §1.5(ii)
Permalink:: http://dlmf.nist.gov/28.32

§28.32(i) Elliptical Coordinates and an Integral Relationship
§28.32(ii) Paraboloidal Coordinates

§28.32(i) Elliptical Coordinates and an Integral Relationship

Notes:: See Arscott (1964b, §§1.3 and 2.6).
Keywords:: Hill’s equation, Mathieu functions, coordinate systems, elliptical coordinates, modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.32.i

If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. These are given by

28.32.1

$x=c\mathop{\cosh\/}\nolimits\xi\mathop{\cos\/}\nolimits\eta,$

$y=c\mathop{\sinh\/}\nolimits\xi\mathop{\sin\/}\nolimits\eta.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : real variable, $\eta$ : variable and $\xi$ : variable
Referenced by:: §28.33(ii)
Permalink:: http://dlmf.nist.gov/28.32.E1
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The two-dimensional wave equation

28.32.2 $\frac{{\partial}^{2}V}{{\partial x}^{2}}+\frac{{\partial}^{2}V}{{\partial y}^{% 2}}+k^{2}V=0$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real variable, : real variable, : parameter and $V(\xi,\eta)$ : solution
Referenced by:: §28.33(ii)
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then becomes

28.32.3 $\frac{{\partial}^{2}V}{{\partial\xi}^{2}}+\frac{{\partial}^{2}V}{{\partial\eta% }^{2}}+\frac{1}{2}c^{2}k^{2}(\mathop{\cosh\/}\nolimits\!\left(2\xi\right)-% \mathop{\cos\/}\nolimits\!\left(2\eta\right))V=0.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\eta$ : variable, $\xi$ : variable, : parameter and $V(\xi,\eta)$ : solution
Referenced by:: §28.32(i), §28.33(ii)
Permalink:: http://dlmf.nist.gov/28.32.E3
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The separated solutions $V(\xi,\eta)=v(\xi)w(\eta)$ can be obtained from the modified Mathieu’s equation (28.20.1) for and from Mathieu’s equation (28.2.1) for , where is the separation constant and $q=\tfrac{1}{4}c^{2}k^{2}$ .

This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta=i\xi$ , $z=\eta$ in (28.32.3)).

Let $u(\zeta)$ be a solution of Mathieu’s equation (28.2.1) and $K(z,\zeta)$ be a solution of

28.32.4 $\frac{{\partial}^{2}K}{{\partial z}^{2}}-\frac{{\partial}^{2}K}{{\partial\zeta% }^{2}}=2q\left(\mathop{\cos\/}\nolimits\!\left(2z\right)-\mathop{\cos\/}% \nolimits\!\left(2\zeta\right)\right)K.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $q=h^{2}$ : parameter, : complex variable, $\zeta$ : variable and $K(z,\zeta)$ : solution
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Also let $\mathcal{L}$ be a curve (possibly improper) such that the quantity

28.32.5 $K(z,\zeta)\frac{du(\zeta)}{d\zeta}-u(\zeta)\frac{\partial K(z,\zeta)}{\partial\zeta}$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : complex variable, $\zeta$ : variable, $u(\zeta)$ : solution and $K(z,\zeta)$ : solution
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approaches the same value when $\zeta$ tends to the endpoints of $\mathcal{L}$ . Then

28.32.6 $w(z)=\int_{{\mathcal{L}}}K(z,\zeta)u(\zeta)d\zeta$

Symbols:: : differential of , $\int$ : integral, : complex variable, $\zeta$ : variable, $u(\zeta)$ : solution, $K(z,\zeta)$ : solution and $\mathcal{L}$ : curve
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defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to uniformly on compact subsets of $\Complex$ .

Kernels can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. See Schmidt and Wolf (1979).

§28.32(ii) Paraboloidal Coordinates

Notes:: See Arscott (1964b, p. 23).
Keywords:: Helmholtz equation, Hill’s equation, Mathieu functions, Whittaker–Hill equation, coordinate systems, paraboloidal coordinates
Permalink:: http://dlmf.nist.gov/28.32.ii

The general paraboloidal coordinate system is linked with Cartesian coordinates via

28.32.7

$x_{1}=\tfrac{1}{2}c\left(\mathop{\cosh\/}\nolimits\!\left(2\alpha\right)+% \mathop{\cos\/}\nolimits\!\left(2\beta\right)-\mathop{\cosh\/}\nolimits\!\left% (2\gamma\right)\right),$

$x_{2}=2c\mathop{\cosh\/}\nolimits\alpha\mathop{\cos\/}\nolimits\beta\mathop{% \sinh\/}\nolimits\gamma,$

$x_{3}=2c\mathop{\sinh\/}\nolimits\alpha\mathop{\sin\/}\nolimits\beta\mathop{% \cosh\/}\nolimits\gamma,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable and : parameter
Permalink:: http://dlmf.nist.gov/28.32.E7
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where is a parameter, $0\leq\alpha<\infty$ , $-\pi<\beta\leq\pi$ , and $0\leq\gamma<\infty$ . When the Helmholtz equation

28.32.8 $\nabla^{2}V+k^{2}V=0$

Symbols:: : parameter
Permalink:: http://dlmf.nist.gov/28.32.E8
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is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which are separation constants. Two conditions are used to determine . The first is the $2\pi$ -periodicity of the solutions; the second can be their asymptotic form. For further information see Arscott (1967) for $k^{2}<0$ , and Urwin and Arscott (1970) for $k^{2}>0$ .

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.31 Equations of Whittaker–Hill and Ince 28.33 Physical Applications

§28.32 Mathematical Applications

Contents

§28.32(i) Elliptical Coordinates and an Integral Relationship

§28.32(ii) Paraboloidal Coordinates