 However, we do get a unique extension of the factorial function to the non-integers by insisting that this equation continue to hold when the arbitrary integer formula_39 is replaced by an arbitrary complex number formula_31.Multiplying both sides by formula_30 gives
This infinite product converges for all complex numbers formula_31 except the negative integers, which fail because trying to use the recursion relation formula_48 backwards through the value formula_49 involves a division by zero.Similarly for the gamma function, the definition as an infinite product due to Euler is valid for all complex numbers formula_31 except the non-positive integers