The Gamma Function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers.
For any positive integer, n : Γ(z) = (n-1)!

Derived by Daniel Bernoulli, for complex numbers with a positive real part the gamma function is defined via a convergent improper integral:

Γ(z) = ʃx^(z-1)*1/e^x dx from 0 to infinity.


The gamma function can be seen as a solution to the following interpolation problem:
"Find a smooth curve that connects the points (x,y) given by (x-1)! at the positive integer values for x.”
A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of x. The simple formula for the factorial, x!=1x2x...x X, cannot be used directly for fractional values of x since it is only valid when x is a natural number (or positive integer).
There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express x!; but it is possible to find a general formula for factorials using tools such as integrals and limits from calculus. A good solution to this is the gamma function.