the gamma function
/ GAM-uh /
The factorial n! = 1 * 2 * 3 * ... * n is one of the friendliest things in mathematics, but it only knows whole numbers. It is meaningless to ask for (1/2)! or pi! straight off. The gamma function, written Gamma(z), is the answer to a natural longing: is there a smooth, single function defined for (almost) every complex number z that agrees with the factorial at the integers? There is, and it is essentially unique. Gamma is the continuous, complex-analytic extension of the factorial — the factorial 'grown up' so it can take any value in the plane.
The link to the factorial is a small shift: Gamma(n) = (n-1)! for every positive integer n. So Gamma(1) = 0! = 1, Gamma(2) = 1! = 1, Gamma(3) = 2! = 2, Gamma(4) = 3! = 6, and so on. (The off-by-one shift is a historical inheritance from Euler's integral.) The defining picture for Re z > 0 is Euler's integral, Gamma(z) = integral from 0 to infinity of t^(z-1) e^(-t) dt, and from there the function is continued to the whole plane. The result is holomorphic everywhere except for simple poles at z = 0, -1, -2, -3, ..., the non-positive integers; everywhere else it is a perfectly smooth analytic function.
Gamma matters because it is the single most ubiquitous special function after the elementary ones. It runs through the beta function, the volume of high-dimensional balls, the normalisation of probability distributions (the gamma, chi-squared and Student distributions all carry its name or its values), the functional equation of the Riemann zeta function, and countless integral evaluations. A common surprise: although Gamma extends the factorial, it does NOT extend it 'in only one possible way' unless you also demand log-convexity (the Bohr-Mollerup theorem) — there are other smooth interpolations, but Gamma is the one that is analytic and well behaved, so it is the right one.
The single most quoted special value is Gamma(1/2) = sqrt(pi). It comes from Euler's integral: Gamma(1/2) = integral from 0 to infinity of t^(-1/2) e^(-t) dt, and the substitution t = u^2 turns it into 2 * integral from 0 to infinity of e^(-u^2) du = sqrt(pi), the famous Gaussian integral. From this and the functional equation, Gamma(3/2) = (1/2) Gamma(1/2) = sqrt(pi)/2, and Gamma(5/2) = (3/2)(1/2) sqrt(pi) = 3 sqrt(pi)/4.
Gamma(1/2) = sqrt(pi) is the half-integer factorial; it is why sqrt(pi) appears in the volumes of spheres in odd dimensions.
Gamma is never zero anywhere in the plane — it has poles but no zeros. So 1/Gamma(z) is an entire function (holomorphic everywhere), with zeros exactly at the non-positive integers; that reciprocal is often the cleaner object to work with.