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Adding short description: "Mathematical function"

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{{Short description|Mathematical function}}
In [[mathematics]], '''conical functions''' or '''Mehler functions''' are [[function (mathematics)|functions]] which can be expressed in terms of [[Legendre function]]s of the first and second kind,
<math>P^\mu_{-(1/2)+i\lambda}(x)</math> and <math>Q^\mu_{-(1/2)+i\lambda}(x).</math>

The functions <math>P^\mu_{-(1/2)+i\lambda}(x)</math> were introduced by [[Gustav Ferdinand Mehler]], in 1868, when expanding in series the distance of a point on the axis of a cone to a point located on the surface of the cone. Mehler used the notation <math>K^\mu(x)</math> to represent these functions. He obtained integral representation and series of functions representations for them. He also established an addition theorem
for the conical functions. [[Carl Neumann]] obtained an expansion of the functions <math>K^\mu(x)</math> in terms
of the [[Legendre polynomials]] in 1881. Leonhardt introduced for the conical functions the equivalent of the [[spherical harmonics]] in 1882.

==External links==
*{{dlmf|first=T. M. |last=Dunster|id=14.20|title=Conical (or Mehler) Functions}}
* G. F. Mehler "[http://www.digizeitschriften.de/resolveppn/GDZPPN002153386 Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper]" ''Journal für die reine und angewandte Mathematik'' '''68''', 134 (1868).
* G. F. Mehler "[http://www.digizeitschriften.de/resolveppn/GDZPPN002246112 Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung]" ''Mathematische Annalen'' '''18''' p. 161 (1881).
* C. Neumann "[http://www.digizeitschriften.de/resolveppn/GDZPPN002246120 Ueber die Mehler'schen Kegelfunctionen und deren Anwendung auf elektrostatische Probleme]" ''Mathematische Annalen'' '''18''' p. 195 (1881).
* G. Leonhardt "[http://www.digizeitschriften.de/resolveppn/GDZPPN002246694 Integraleigenschaften der adjungirten Kegelfunctionen]" ''Mathematische Annalen'' '''19''' p. 578 (1882).
* {{MathWorld|title=Conical function|urlname=ConicalFunction}}
* Milton Abramowitz and Irene Stegun (Eds.) ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'' (Dover, 1972) [http://www.math.sfu.ca/~cbm/aands/page_337.htm p. 337]
* A. Gil, J. Segura, N. M. Temme "[http://oai.cwi.nl/oai/asset/13868/13868A.pdf Computing the conical function $P^{\mu}_{-1/2+i\tau}(x)$]" ''SIAM J. Sci. Comput.'' 31(3), 1716–1741 (2009).
* Tiwari, U. N.; Pandey, J. N. The Mehler-Fock transform of distributions. ''Rocky Mountain J. Math.'' 10 (1980), no. 2, 401–408.

[[Category:Special functions]]

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Conical function - Revision history

imported>Sebrana: Adding short description: "Mathematical function"