Whittaker function
Encyclopedia
In mathematics, a Whittaker function is a special solution of Whittaker
's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Whittaker functions of reductive group
s over local field
s, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R).
Whittaker's equation is
It has a regular singular point at 0 and an irregular singular point at ∞.
Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by
Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker model
s.
E. T. Whittaker
Edmund Taylor Whittaker FRS FRSE was an English mathematician who contributed widely to applied mathematics, mathematical physics and the theory of special functions. He had a particular interest in numerical analysis, but also worked on celestial mechanics and the history of physics...
's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Whittaker functions of reductive group
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
s over local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R).
Whittaker's equation is
It has a regular singular point at 0 and an irregular singular point at ∞.
Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by
Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker model
Whittaker model
In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space of functions on the group. It is named after E. T...
s.