Gamma Function

The gamma function extends the factorial operation, ordinarily defined only for non-negative whole numbers, to almost the entire complex plane, satisfying the same defining relationship as the factorial, that the gamma function at n plus one equals n times the gamma function at n, while also being defined and smoothly varying at non-integer and negative non-integer arguments where an ordinary factorial has no meaning. It was introduced by the Swiss mathematician Leonhard Euler in the eighteenth century, who first expressed it as an infinite product before Adrien-Marie Legendre later introduced the integral representation and the now-standard capital-gamma notation still used today. The gamma function has poles, meaning points where it becomes infinite, at zero and at every negative whole number, but is otherwise defined everywhere else in the complex plane, and it satisfies a number of classical identities, including a reflection formula relating its values at a point and at one minus that point, and Legendre's duplication formula relating its value at a point to its values at half that point and half that point plus one half. Because it interpolates the factorial so naturally, the gamma function appears throughout probability theory, where it underlies the formulas for the gamma and beta probability distributions, as well as in analytic number theory, particularly in the study of the Riemann zeta function's own functional equation.