Define a Gamma function

The Gamma function is a special function that extends the factorial function to complex numbers. It is denoted by the Greek letter Γ.

The Gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n, Γ(n) = (n-1)!.

The general definition of the Gamma function for complex numbers z (where Re(z) > 0) is given by an improper integral:

Γ(z) = ∫ from 0 to ∞ of t^(z-1)e^(-t) dt

This integral converges when the real part of z is greater than zero.

The Gamma function has many interesting properties, including the recursive property Γ(z+1) = zΓ(z), which is similar to the property n! = n*(n-1)! for factorials.

The Gamma function is used in many areas of mathematics, including calculus, complex analysis, and number theory.