GammaFunktion

The Gamma function, denoted by the Greek letter Gamma (Γ), is a special function in mathematics that extends the factorial function to complex numbers. It is defined for all complex numbers except the non-positive integers, where it has simple poles. The Gamma function is defined by the improper integral:

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

where z is a complex number with a positive real part. This integral converges for all complex numbers z except for z = 0, -1, -2, etc., where the Gamma function has simple poles.

The Gamma function has several important properties, including:

1. Recurrence relation: Γ(z+1) = z * Γ(z)

2. Reflection formula: Γ(z) * Γ(1-z) = π / sin(πz)

3. Functional equation: Γ(z) = (z-1)!

The Gamma function is widely used in various areas of mathematics and physics, such as calculus, number theory, and quantum mechanics. It appears in the definition of the Riemann zeta function, the partition function in statistical mechanics, and the wave function in quantum mechanics. The Gamma function is also used to define the Beta function, which is another important special function in mathematics.