# Generalized spheroidal wave equation and limiting cases

## Abstract

We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a double-confluent Heun equation by a taking-limit process due to Leaver. [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. Another procedure, called Whittaker-Ince limit [B. D. Figueiredo, J. Math. Phys. 46, 113503 (2005)], provides solutions in series of powers and Bessel functions for two other equations with a different type of singularity at infinity. In addition, new solutions are obtained for the Whittaker-Hill and Mathieu equations [F. M. Arscott, Proc. R. Soc. Edinburg A67, 265 (1967)] by considering these as special cases of both the confluent and double-confluent Heun equations. In particular, we find that each of the Lindemann-Stieltjes solutions for the Mathieu equation [E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Pressmore »

- Authors:

- Instituto de Cosmologia, Relatividade e Astrofisica (ICRA-BR), Centro Brasileiro de Pesquisas Fisicas (CBPF), Rua Dr. Xavier Sigaud, CEP 22290-180, Rio de Janeiro 150 (Brazil)

- Publication Date:

- OSTI Identifier:
- 20929619

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2406057; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMMETRY; BESSEL FUNCTIONS; GEOMETRY; HYPERGEOMETRIC FUNCTIONS; MATHEMATICAL SOLUTIONS; MATHIEU EQUATION; MORSE POTENTIAL; SCHROEDINGER EQUATION; SINGULARITY

### Citation Formats

```
Bonorino Figueiredo, B. D.
```*Generalized spheroidal wave equation and limiting cases*. United States: N. p., 2007.
Web. doi:10.1063/1.2406057.

```
Bonorino Figueiredo, B. D.
```*Generalized spheroidal wave equation and limiting cases*. United States. doi:10.1063/1.2406057.

```
Bonorino Figueiredo, B. D. Mon .
"Generalized spheroidal wave equation and limiting cases". United States.
doi:10.1063/1.2406057.
```

```
@article{osti_20929619,
```

title = {Generalized spheroidal wave equation and limiting cases},

author = {Bonorino Figueiredo, B. D.},

abstractNote = {We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a double-confluent Heun equation by a taking-limit process due to Leaver. [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. Another procedure, called Whittaker-Ince limit [B. D. Figueiredo, J. Math. Phys. 46, 113503 (2005)], provides solutions in series of powers and Bessel functions for two other equations with a different type of singularity at infinity. In addition, new solutions are obtained for the Whittaker-Hill and Mathieu equations [F. M. Arscott, Proc. R. Soc. Edinburg A67, 265 (1967)] by considering these as special cases of both the confluent and double-confluent Heun equations. In particular, we find that each of the Lindemann-Stieltjes solutions for the Mathieu equation [E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press (1945)] is associated with two expansions in series of Bessel functions. We also discuss a set of solutions in series of hypergeometric and confluent hypergeometric functions for the GSWE and use their Leaver limits to obtain infinite-series solutions for the Schroedinger equation with an asymmetric double-Morse potential. Finally, the possibility of extending the solutions of the GSWE to the general Heun equation is briefly discussed.},

doi = {10.1063/1.2406057},

journal = {Journal of Mathematical Physics},

number = 1,

volume = 48,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}