In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, conical functions or Mehler functions are
functions which can be expressed in terms of
Legendre function
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated L ...
s of the first and second kind,
and
The functions
were introduced by
Gustav Ferdinand Mehler
Gustav Ferdinand Mehler, or Ferdinand Gustav Mehler (13 December 1835, in Schönlanke, Kingdom of Prussia – 13 July 1895, in Elbing, German Empire) was a German mathematician.
He is credited with introducing Mehler's formula; the Mehler–Foc ...
, 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
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
Carl Gottfried Neumann (also Karl; 7 May 1832 – 27 March 1925) was a German mathematician.
Biography
Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann (1798–1895), who w ...
obtained an expansion of the functions
in terms
of the
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
in 1881. Leonhardt introduced for the conical functions the equivalent of the
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
in 1882.
External links
*
* G. F. Mehler
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
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
Ueber die Mehler'schen Kegelfunctionen und deren Anwendung auf elektrostatische Probleme ''Mathematische Annalen'' 18 p. 195 (1881).
* G. Leonhardt
Integraleigenschaften der adjungirten Kegelfunctionen ''Mathematische Annalen'' 19 p. 578 (1882).
*
* Milton Abramowitz and Irene Stegun (Eds.) ''
Handbook of Mathematical Functions'' (Dover, 1972
p. 337* A. Gil, J. Segura, N. M. Temme
Computing the conical function $P^_(x)$ ''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.
Special functions
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