In mathematics, the gamma function 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 except non-positive integers, and for every positive integer . The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: 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.