Gamma function

In 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 integer ⁠ n {\displaystyle n} ⁠. The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t , 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 reciprocal 1 / 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).} Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It appears as a factor in various probability-distribution functions and other formulas in the fields of probability, statistics, analytic number theory, and combinatorics.