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:
or in the following equivalent form:
By a change of variables the integral becomes the following form:
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.
For integer values we have:
As generell formular we have:
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:
First derivative of the gamma function:
N'th derivatives of the gamma function:
Relation between the Riemann zeta function and the gamma function