In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 ...
s of the first and second kind,
and
The functions
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
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 Mathematical physics, mathematical physicist and professor at several German universities. His work focused on applications of potential theory to physics and mathemati ...
obtained an expansion of the functions
in terms
of the
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
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. The table of spherical harmonics co ...
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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