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Type 0.5! in your calculator to see what the factorial of one-half is. The result will be 0.886..., and the exact answer is the square root of pi divided by 2--amazing! How is this possible, when the factorial of a number n is defined as n! = n(n-1)(n-2)...1 and this definition only makes sense for whole numbers?
The calculator result is not an error, and in this video I explain how the factorial can be extended beyond the whole numbers for all real numbers by the gamma function. Once we extend the factorial function beyond whole numbers, you can see why the factorial of one-half is equal to the square root of pi divided by 2.
Bohr-Mollerup theorem
http://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem
Applications
https://www.math.washington.edu/~morrow/336_10/papers/joel.pdf
Numerical computation
http://www.rskey.org/CMS/index.php/the-library/11
Alternative ways to extend the factorial function
http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html
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