The function itself was initially defined as an integral (in modern representation, ) for positive real x by Swiss mathematician Leonhard Euler in 1730. The notation Γ(x) was introduced by Adrien-Marie Legendre. Both Euler's integral and Legendre's notation shift the argument with respect to the factorial, so that for integer n>0, Γ(n) = (n−1)!. Carl Friedrich Gauss preferred π(x), with no shift, but Legendre's notation prevailed. Generalisation to non-integer negative and to complex numbers was achieved by analytic continuation.
- (mathematics, analysis) A meromorphic function which generalises the notion of factorial to complex numbers and has singularities at the nonpositive integers.