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