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

§28.32 Mathematical Applications

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Keywords:: Mathieu functions, applications, mathematical
Referenced by:: §1.5(ii)
Permalink:: http://dlmf.nist.gov/28.32
See also:: Annotations for Ch.28

§28.32(i) Elliptical Coordinates and an Integral Relationship

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Keywords:: Hill’s equation, Mathieu functions, applications, coordinate systems, elliptical, elliptical coordinates, mathematical, modified Mathieu functions, separation constants
Notes:: See Arscott (1964b, §§1.3 and 2.6).
Permalink:: http://dlmf.nist.gov/28.32.i
See also:: Annotations for §28.32 and Ch.28

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	$\displaystyle x$	$\displaystyle=c\cosh\xi\cos\eta,$
28.32.1	$\displaystyle y$	$\displaystyle=c\sinh\xi\sin\eta.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable, $\eta$ : variable and $\xi$ : variable Referenced by: §28.33(ii) Permalink: http://dlmf.nist.gov/28.32.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.32(i), §28.32 and Ch.28

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: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $x$ : real variable, $y$ : real variable, $k$ : parameter and $V(\xi,\eta)$ : solution Referenced by: §28.33(ii) Permalink: http://dlmf.nist.gov/28.32.E2 Encodings: TeX, pMML, png See also: Annotations for §28.32(i), §28.32 and Ch.28

then becomes

28.32.3		$\frac{{\partial}^{2}V}{{\partial\xi}^{2}}+\frac{{\partial}^{2}V}{{\partial\eta% }^{2}}+\frac{1}{2}c^{2}k^{2}(\cosh\left(2\xi\right)-\cos\left(2\eta\right))V=0.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\eta$ : variable, $\xi$ : variable, $k$ : parameter and $V(\xi,\eta)$ : solution Referenced by: §28.32(i), §28.33(ii) Permalink: http://dlmf.nist.gov/28.32.E3 Encodings: TeX, pMML, png See also: Annotations for §28.32(i), §28.32 and Ch.28

The separated solutions $V(\xi,\eta)=v(\xi)w(\eta)$ can be obtained from the modified Mathieu’s equation (28.20.1) for $v$ and from Mathieu’s equation (28.2.1) for $w$ , where $a$ 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 $\zeta=\mathrm{i}\xi$ , $z=\eta$ 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		$\frac{{\partial}^{2}K}{{\partial z}^{2}}-\frac{{\partial}^{2}K}{{\partial\zeta% }^{2}}=2q\left(\cos\left(2z\right)-\cos\left(2\zeta\right)\right)K.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $q=h^{2}$ : parameter, $z$ : complex variable, $\zeta$ : variable and $K(z,\zeta)$ : solution Permalink: http://dlmf.nist.gov/28.32.E4 Encodings: TeX, pMML, png See also: Annotations for §28.32(i), §28.32 and Ch.28

Also let $\mathcal{L}$ be a curve (possibly improper) such that the quantity

28.32.5		$K(z,\zeta)\frac{\mathrm{d}u(\zeta)}{\mathrm{d}\zeta}-u(\zeta)\frac{\partial K(% z,\zeta)}{\partial\zeta}$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : complex variable, $\zeta$ : variable, $u(\zeta)$ : solution and $K(z,\zeta)$ : solution Permalink: http://dlmf.nist.gov/28.32.E5 Encodings: TeX, pMML, png See also: Annotations for §28.32(i), §28.32 and Ch.28

approaches the same value when $\zeta$ tends to the endpoints of $\mathcal{L}$ . Then

28.32.6		$w(z)=\int_{\mathcal{L}}K(z,\zeta)u(\zeta)\,\mathrm{d}\zeta$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $\zeta$ : variable, $u(\zeta)$ : solution, $K(z,\zeta)$ : solution and $\mathcal{L}$ : curve Permalink: http://dlmf.nist.gov/28.32.E6 Encodings: TeX, pMML, png See also: Annotations for §28.32(i), §28.32 and Ch.28

defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $\mathbb{C}$ .

Kernels $K$ 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

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Keywords:: Helmholtz equation, Hill’s equation, Mathieu functions, Whittaker–Hill equation, applications, coordinate systems, mathematical, paraboloidal, paraboloidal coordinates, separation constants
Notes:: See Arscott (1964b, p. 23).
Permalink:: http://dlmf.nist.gov/28.32.ii
See also:: Annotations for §28.32 and Ch.28

The general paraboloidal coordinate system is linked with Cartesian coordinates via

28.32.7	$\displaystyle x_{1}$	$\displaystyle=\tfrac{1}{2}c\left(\cosh\left(2\alpha\right)+\cos\left(2\beta% \right)-\cosh\left(2\gamma\right)\right),$
	$\displaystyle x_{2}$	$\displaystyle=2c\cosh\alpha\cos\beta\sinh\gamma,$
	$\displaystyle x_{3}$	$\displaystyle=2c\sinh\alpha\sin\beta\cosh\gamma,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : sine function, $x$ : real variable and $c$ : parameter Permalink: http://dlmf.nist.gov/28.32.E7 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §28.32(ii), §28.32 and Ch.28

where $c$ 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: $k$ : parameter Permalink: http://dlmf.nist.gov/28.32.E8 Encodings: TeX, pMML, png See also: Annotations for §28.32(ii), §28.32 and Ch.28

is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which $A, B$ are separation constants. Two conditions are used to determine $A, B$ . 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 $k^{2}>0$ .

28.31 Equations of Whittaker–Hill and Ince 28.33 Physical Applications

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§28.32 Mathematical Applications

Contents

§28.32(i) Elliptical Coordinates and an Integral Relationship

§28.32(ii) Paraboloidal Coordinates

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