Gamma Calcs

Asked myself the question, if I wanted to find the
factorial to a fraction, what would I need to do;
in other words, what do the values of the Gamma
function represent. Turns out to be a difficult question.

Perhaps not for Gauss, who established that the factorial
of 1/2 is the square root of pi. Not bad, and values for
other fractions start there.

source: (Russian)
 Wikipedia

Tested the formula for (1/2 + 2)

The 'gaussian integral' was confirmed by mathematicians who came after
him. The proof uses double integration, with one taking on the parameters
of the complex plane.

https://en.wikipedia.org/wiki/Gaussian_integral

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It's like taking yourself out on a date - der Integration durch Substitution auf Funktionen höherer Dimensionen - lets one solve difficult integration problem by moving to higher dimensions.
So minus infinity to positive infinity happens on the one dimension of real numbers, whereas
the complex plane adds another dimension, allowing one to solve 0 to 2pi!!

https://de.wikipedia.org/wiki/Transformationssatz