Confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term confluent refers to the merging of singular points of families of differential equations; confluere is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: Kummer's (confluent hypergeometric) function M(a, b, z), introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. Tricomi's (confluent hypergeometric) function U(a, b, z) introduced by , sometimes denoted by Ψ(a; b; z), is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. Whittaker functions (for Edmund Taylor Whittaker) are solutions to Whittaker's equation. Coulomb wave functions are solutions to the Coulomb wave equation. The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables. Kummer's equation may be written as: with a regular singular point at z = 0 and an irregular singular point at z = ∞. It has two (usually) linearly independent solutions M(a, b, z) and U(a, b, z). Kummer's function of the first kind M is a generalized hypergeometric series introduced in , given by: where: is the rising factorial. Another common notation for this solution is Φ(a, b, z). Considered as a function of a, b, or z with the other two held constant, this defines an entire function of a or z, except when b = 0, −1, −2, ... As a function of b it is analytic except for poles at the non-positive integers. Some values of a and b yield solutions that can be expressed in terms of other known functions. See #Special cases.