Heun Functions

Heun functions are generalizations of hypergeometric functions that occur in quantum mechanics, mathematical physics and other applications. There are a variety of Heun function types. A number of these are summarized in the following table together with their implementations in the Wolfram Language.

type	function	derivate function
general	HeunG	HeunGPrime
confluent	HeunC	HeunCPrime
double confluent	HeunD	HeunDPrime
bi-confluent	HeunB	HeunBPrime
tri-confluent	HeunT	HeunTPrime
Lamé	LameC	LameCPrime
Lamé	LameS	LameSPrime

satisfies the general Heun differential equation

(1)

satisfies the confluent Heun differential equation

(2)

satisfies the double-confluent Heun differential equation

(3)

satisfies the bi-confluent Heun differential equation

(4)

satisfies the tri-confluent Heun differential equation

(5)

satisfies the Lam' differential equation

(6)

and satisfies the Lamé differential equation

(7)

where in the latter two, is a Jacobi elliptic function with elliptic modulus .