Conical function
Encyclopedia
In mathematics
, conical functions or Mehler functions are functions
which can be expressed in terms of Legendre function
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
obtained an expansion of the functions
in terms
of the Legendre polynomials in 1881. Leonhardt introduced for the conical functions the equivalent of the spherical harmonics
in 1882.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, conical functions or Mehler functions are functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
which can be expressed in terms of Legendre function
Legendre function
In mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions P, Q are generalizations of Legendre polynomials to non-integer degree.-Differential equation:...
s of the first and second kind,
and The functions
were introduced by Gustav Ferdinand MehlerGustav Ferdinand Mehler
Gustav Ferdinand Mehler, or Ferdinand Gustav Mehler was a German mathematician, who introduced Mehler's formula, Mehler–Fock formula, Mehler–Heine formula, Mehler functions....
, 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 theoremfor the conical functions. Carl Neumann
Carl Neumann
Carl Gottfried Neumann was a German mathematician.Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann , who was professor of mineralogy and physics at Königsberg University...
obtained an expansion of the functions
in termsof the Legendre polynomials in 1881. Leonhardt introduced for the conical functions the equivalent of the spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
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 FunctionsHandbook of mathematical functions*Abramowitz and Stegun, the informal name for Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*NIST Handbook of Mathematical Functions...
(Dover, 1972) p. 337 - A. Gil, J. Segura, N. M. Temme "Computing the conical function $P^{\mu}_{-1/2+i\tau}(x)$" SIAM J. Sci. Comput. 31(3), 1716-1741 (2009).