Gamma function

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The Gamma function is a function which may be used to extend the concept of factorial to complex numbers. If the real part of the complex number z is positive, one can define

Γ ( z ) = tz-1 e-t   dt

and show that

Γ(z+1) = z Γ(z).

Because of Γ(1) = 1, this relation implies Γ(n+1) = n! for all natural numbers n. It can further be used to extend the definition of Γ(z) to all complex numbers z except z = 0, -1, -2, -3, ...