The gamma function along part of the real axis

In mathematics, the gamma function (Γ(z)) is a key topic in the field of special functions. Γ(z) is an extension of the factorial function to all complex numbers except negative integers. For positive integers, it is defined as \displaystyle{ \Gamma(n) = (n-1)! }

The gamma function is defined for all complex numbers, but it is not defined for negative integers and zero. For a complex number whose real part is not a negative integer, the function is defined by:

\displaystyle{ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,{\rm d}t. }