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 , 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 introduced by , sometimes denoted by , 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 Kummer's equation may be written as: with a regular singular point at and an irregular singular point at . It has two (usually) linearly independent solutions and . Kummer's function of the first kind is a generalized hypergeometric series introduced in , given by: where: is the rising factorial. Another common notation for this solution is . Considered as a function of , , or with the other two held constant, this defines an entire function of or , except when As a function of it is analytic except for poles at the non-positive integers. Some values of and yield solutions that can be expressed in terms of other known functions. See #Special cases. When is a non-positive integer, then Kummer's function (if it is defined) is a generalized Laguerre polynomial. Just as the confluent differential equation is a limit of the hypergeometric differential equation as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of the hypergeometric function and many of the properties of the confluent hypergeometric function are limiting cases of properties of the hypergeometric function. Since Kummer's equation is second order there must be another, independent, solution. The indicial equation of the method of Frobenius tells