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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:



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

For x > 0 we have:

Definition Gamma-Function

or in the following equivalent form:

Definition Gamma-Function

By a change of variables the integral becomes the following form:

Definition Gamma-Function

Definition Gamma-Function

Alternative Definition

The gamma function can be defined by using the following product:

Definition Gamma-Function

Relation to the Mellin Transformation

The gamma function Γ(x) is a special case of a Mellin Transformation Mf (x):

Mellin Transformation

The gamma function Γ(x) is the Mellin transform of f(t) = e-t.


For integer values we have:


As generell formular we have:

Gamma function propertie

Negative Values

Rearranging the above formular will give:

Gamma function propertie

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:

gamma function propertie


First derivative of the gamma function:

First derivative of the gamma function

N'th derivatives of the gamma function:

N'th derivative of the gamma function

Relation between the Riemann zeta function and the gamma function

gamma function properties

Further Relations

gamma function sine function

gamma function sine function

gamma function sine function

Special Values

gamma function sine function

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27. August 2017 Version 1.0
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