The gamma function stretches factorials into complex numbers
Written with a capital Greek gamma, the gamma function is the usual way to extend n! beyond the positive integers. Daniel Bernoulli studied it; defined for all complex numbers except non-positive integers, it continues an integral formula across the plane as a meromorphic function with simple poles.
The problem it solves is interpolation: find a smooth curve passing through every point of the factorial sequence 1, 2, 6, 24 and onward. No elementary function manages it. The gamma function does, and it is not merely smooth but analytic everywhere except zero and the negative integers. It is also the Mellin transform of the decaying exponential e to the minus x, which is one reason it turns up in probability distributions and number theory.
Its classic definition integrates t raised to z minus 1 times e to the minus t from zero to infinity, which converges absolutely whenever z has positive real part. Euler's name is attached twice: this is his integral of the second kind, while his integral of the first kind gives the beta function. Evaluating at z equal to 1 yields exactly 1, and integrating by parts shows that gamma of z plus 1 equals z times gamma of z. Applying that rule over and over reproduces the factorial at whole numbers and pushes the definition leftward across the plane.
Uniqueness is subtler than it looks. Adding any analytic function that vanishes at the positive integers produces another curve through the same factorial points; such impostors are called pseudogamma functions, the Hadamard function being the best known. Even the shift rule allows multiplication by periodic factors. The Bohr–Mollerup theorem settles the matter: on the positive reals, gamma is the only function obeying the shift rule, equal to 1 at 1, whose logarithm is convex. The capital-gamma notation itself goes back to Legendre.
Euler also found a second route. Assuming that n factorial times n plus 1 to the power z, divided by n plus z factorial, tends to 1, he built an infinite product for gamma that converges for every complex number except the non-positive integers, where a division by zero wrecks it.
Source: Gamma function
