Intuition behind $(-\frac{1}{2})! = \sqrt{\pi}$ It can be shown that using the definition of the Gamma function as:
$$\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} dx $$ that $$\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$$ or slightly abusing notation, that $(-\frac{1}{2})! = \sqrt{\pi}$. Is there an intuitive explanation to this?
I want to make clear that I am not per se interested in a proof of this fact (most often these are clever technical manipulations) but in insight into this phenomenon.  


Is there an intuitive explanation to this ?

Yes. There is an umbilical connection between $\bigg(\dfrac1n\bigg){\large!}$ or $\Gamma\bigg(\dfrac1n\bigg)$, and geometric shapes  of the form $X^n+Y^n=R^n\iff x^n+y^n=1\iff y=\sqrt[n]{1-x^n}$, whose area is  $\displaystyle\int_0^1\sqrt[n]{1-x^n}~dx$, which is nothing else than the beta function in disguise. Ultimately, it's  all related to Newton's binomial theorem. The latter expands the power of a sum into a  sum of powers, with the help of binomial coefficients, or beta functions, which are then  expressed in terms of factorials, or $\Gamma$ functions.
