the gamma function
The factorial n! multiplies the whole numbers together, but only makes sense for whole numbers. The gamma function is its smooth completion: a single continuous curve that passes exactly through the factorials yet is defined for fractions, negatives, and even complex numbers. It answers 'what is (1/2)! ?' -- and the answer, remarkably, involves sqrt(pi).
For Re(z) greater than 0 the gamma function is Gamma(z) = the integral from 0 to infinity of t^(z-1) e^(-t) dt. Its defining property is the recursion Gamma(z + 1) = z Gamma(z), which with Gamma(1) = 1 forces Gamma(n + 1) = n! for non-negative integers n. The integral converges only for Re(z) greater than 0, but the recursion analytically continues Gamma to the whole complex plane, where it is holomorphic except for simple poles at z = 0, -1, -2, ... (the residue at z = -n is (-1)^n / n!). A key special value is Gamma(1/2) = sqrt(pi).
It is the connective tissue of special functions and statistics: it fixes the normalization of the Gaussian and of the beta, gamma, chi-squared, and Student distributions, appears in the volume of an n-dimensional sphere (hence in the phase-space counting of statistical mechanics), and defines the analytic-continuation tricks (dimensional regularization) that tame divergent integrals in quantum field theory. Stirling's approximation, Gamma(z + 1) approx sqrt(2 pi z) (z/e)^z, is the everyday large-n factorial estimate behind Boltzmann's entropy counting.
Gamma(1/2) = sqrt(pi) is the Gaussian integral in disguise: the integral of e^(-t)/sqrt(t) from 0 to infinity = sqrt(pi). It is why a half-integer factorial carries a sqrt(pi).
The gamma function assigns (-1/2)! = sqrt(pi).
The gamma function has no zeros anywhere, but it does have poles at zero and the negative integers; its reciprocal 1/Gamma(z) is the entire function, vanishing there. Do not expect Gamma to be finite at a non-positive integer.