There are lots of analytic functions with this property (and agreeing with the ordinary gamma function on integer arguments). For example, gamma(x) * cos(2πx) [

There are lots of analytic functions with this property (and agreeing with the ordinary gamma function on integer arguments). For example, gamma(x) * cos(2πx) [or, in complete generality, gamma(x) * f(x) for any analytic function f of period 1, normalized to take value 1 at integers; thus, there are infinitely many examples]. Rather, the gamma function is the unique function satisfying this recurrence, taking the usual value at integers, with the asymptotics that gamma(n + x)/(gamma(n) * n^x) approaches 1 as x is held fixed and n grows large. [See http://www.quora.com/How-exactly-does-the-gamma-function-extend-the-factorial-function-to-the-reals/answer/Sridhar-Ramesh http://www.quora.com/How-exactly-does-the-gamma-function-ext... for a detailed exposition]