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 M_f (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

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