Prove that #((1/2)!)=sqrt(pi)/2#?
1 Answer
We are going to begin by switching to a much more useful form: the Gamma function, which is defined thus:
The last property technically means that this is an analytical continuation/extension of the factorial function, which arguably only makes sense on integers.
Anyway, by the above,
This is a bit difficult (it involves three u-substitutions versus our 2), so let's apply another property of factorial:
hence
While this seems more complicated, consider the
We can do the Gaussian integral by hand (or you can just know its value):
Therefore,