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### Gamma function

The gamma function, indicated with the Greek letter Γ, is an extension of the factorial function to complex numbers, and is defined as

##### Explanation

This improper integral has the important property that Г(* p*) = (* p* – 1) !, in which *p* is an integer greater than or equal to 1.

Actually, the factorial function is a special case of the gamma function, because

for all natural numbers *n*.

##### Properties

1. For *p* = 1

2. Replacing *t = x*^{2} ⇒ *dt* = 2*x dx*

then now for

##### Example 1

The factorial of the transcendental number π can be calculated as

##### Example 2

The factorial of the transcendental number *e* can be calculated as

##### Example 3

The factorial of the imaginary unit *i* can be calculated as

i! = Γ(1 +i) ≈ 0,4980 − 0,1549i

##### Example 4

The gamma function is used in the power series for the inverse cosine

##### Additional information

The gamma function was described by the Swiss mathematician Leonhard Euler in 1729.