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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 = x2   ⇒   dt = 2x 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.

 


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