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 Kummer (1837), is a solution to Kummer's differential equation. There is a different and unrelated Kummer's function bearing the same name.
• Tricomi's (confluent hypergeometric) function U(a;b;z) introduced by Francesco Tricomi (1947), sometimes denoted by Ψ(a;b;.z), is another solution to Kummer's equation.
• 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.