# §28.32 Mathematical Applications

## §28.32(i) Elliptical Coordinates and an Integral Relationship

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 $\displaystyle x$ $\displaystyle=c\cosh\xi\cos\eta,$ $\displaystyle y$ $\displaystyle=c\sinh\xi\sin\eta.$

The two-dimensional wave equation

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

then becomes

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

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

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

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

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

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

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

The general paraboloidal coordinate system is linked with Cartesian coordinates via

 28.32.7 $\displaystyle x_{1}$ $\displaystyle=\tfrac{1}{2}c\left(\cosh\left(2\alpha\right)+\cos\left(2\beta% \right)-\cosh\left(2\gamma\right)\right),$ $\displaystyle x_{2}$ $\displaystyle=2c\cosh\alpha\cos\beta\sinh\gamma,$ $\displaystyle x_{3}$ $\displaystyle=2c\sinh\alpha\sin\beta\cosh\gamma,$

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 $k^{2}>0$.