 Confluent hypergeometric function

Confluent hypergeometric functions solve the Kummer differential equation $z{y}^{\prime \prime }+(b-z)y^{\prime }-ay=0$. The regular solution is the confluent hypergeometric function of the first kind; a second standard solution is conventionally denoted $U(a,b,z)$. The family arises by coalescing singular points of the Gauss hypergeometric equation.

The Confluent hypergeometric function is a special function that arises in various areas of mathematics and physics, particularly in the context of solving differential equations. It is a limit case of the more general hypergeometric function and is particularly useful in situations where the parameters of the hypergeometric function simplify, leading to the confluent form.