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April 10th, 2015, 05:53 AM   #1
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Question (-0.5)!^2 = π? why?

Why does ((-0.5)!)^2 equal pi? To me it really doesnt make sense since of what i know of you cant even factorial negative numbers right??
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April 10th, 2015, 06:15 AM   #2
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The "standard" definition of factorial is only for non-negative integers but it can be extended. It can, for example, be proven that the gamma function $\displaystyle \Gamma(n)= \int_0^\infty t^{n-1} e^{-t}dt= (n-1)!$ for n any positive integer. Even that integral is not defined for n a negative integer but it can be calculated for half integer, positive or negative, by an "integration by parts". Here "(-0.5)!)" is to be interpreted $\displaystyle \Gamma(0.5)= \int_0^\infty t^{1/2}e^{-t}dt$.
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