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The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
Thus the -gamma function can be considered as an extension of the -factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit = (). There is a simple proof of this limit by Gosper.
For all positive integers, ! = (+), where Γ denotes the gamma function. However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied.
The falling factorial can be extended to real values of using the gamma function provided and + are real numbers that are not negative integers: = (+) (+) , and so can the rising factorial: = (+) . Calculus
One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates the factorials and obeys the same functional equation.
The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on R + at x 0 = 1.461 632 144 968 362 341 26.... All others occur single between the poles on the negative axis:
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation!! = ()!! to give !! = (+)!! +.
The polylogarithm is related to the Hurwitz zeta function by: = () [(, + ()) + (, ())], which relation, however, is invalidated at positive integer s by poles of the gamma function Γ(1 − s), and at s = 0 by a pole of both zeta functions; a derivation of this formula is given under series representations below.