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## Gamma Function - Γ(z)

The Gamma Function - Γ(z) was developed by Leonhard Euler to generalise the factorial n! to non-integer values.

The following relation between the factorial and the gamma function with integer arguments is given: ### Definiton

The first definition of the gamma function was given by Euler in 1730 in a letter to Goldbach.

For x > 0 we have: or in the following equivalent form: By a change of variables the integral becomes the following form:  ### Alternative Definition

The gamma function can be defined by using the following product: ### Relation to the Mellin Transformation

The gamma function Γ(x) is a special case of a Mellin Transformation Mf (x): The gamma function Γ(x) is the Mellin transform of f(t) = e-t.

### Properties

For integer values we have: As generell formular we have: ### Negative Values

Rearranging the above formular will give: The right side of this expression is defined for negative x in the range between -1 > x > 0.

Using the value -1/2 as value for x in this formular we get: For other negative values of x with x+n>0, except negative integers and zero (0, -1, -2, -3, ...) we can generalise the formular above: ### Derivatives

First derivative of the gamma function: N'th derivatives of the gamma function: ### Relation between the Riemann zeta function and the gamma function ### Further Relations   ### Special Values Bild: "Edersee bei Niedrigwasser"

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