Gamma function

Extension of the factorial functionIn mathematics, the gamma function, denoted by ⁠Γ{\displaystyle \Gamma }⁠ (capital Greek letter gamma), is the most common extension of the factorial function to complex numbers. First studied by Daniel Bernoulli, the gamma function is defined for all complex numbers z{\displaystyle z} except non-positive integers, and satisfied Γ(n)=(n−1)!{\displaystyle \Gamma (n)=(n-1)!} for every positive integer⁠n{\displaystyle n}⁠. The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: Γ(z)=∫0∞tz−1e−tdt, ℜ(z)>0.{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt,\ \qquad \Re (z)>0.} The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. Since the gamma function has no zeros, its reciprocal1/Γ{\displaystyle 1/\Gamma } is an entire function. In fact, the gamma function corresponds to the Mellin transform of exponential decay: Γ(z)=M{e−x}(z).{\displaystyle \Gamma (z)={\mathcal {M}}\{e^{-x}\}(z)