# §13.28 Physical Applications

## §13.28(i) Exact Solutions of the Wave Equation

The reduced wave equation $\nabla^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi\eta}\cos\phi$, $y=2\sqrt{\xi\eta}\sin\phi$, $z=\xi-\eta$, can be solved via separation of variables $w=f_{1}(\xi)f_{2}(\eta)e^{\mathrm{i}p\phi}$, where

 13.28.1 $\displaystyle f_{1}(\xi)$ $\displaystyle=\xi^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(1)}(2\mathrm{i}k\xi)$, $\displaystyle f_{2}(\eta)$ $\displaystyle=\eta^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(2)}(-2\mathrm{i}k\eta)$, ⓘ Symbols: $\mathrm{i}$: imaginary unit and $k$: wavenumber Permalink: http://dlmf.nist.gov/13.28.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.28(i), §13.28 and Ch.13

and $V^{(j)}_{\kappa,\mu}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). See Hochstadt (1971, Chapter 7).

For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000).

See Chapter 33.

## §13.28(iii) Other Applications

For dynamics of many-body systems see Meden and Schönhammer (1992); for tomography see D’Ariano et al. (1994); for generalized coherent states see Barut and Girardello (1971); for relativistic cosmology see Crisóstomo et al. (2004).