I have stumbled upon Gamma functions when dealing with Gamma distributions on my studies with basic statistics. However, I have not understood how its computation expands factorials to real and complex numbers through:
$$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt = x!$$
with x as a non-negative integer. Mine is sort of a straightforward question: your input is a non-negative integer and you get the factorials of all real and complex number between the integers as the output? If that's the point, what does it even mean to have the factorial of a real/complex number? Are they arbitrary definitions derived of the Gamma function?
P.S.: I have searched the forum and found some interesting discussions on the topic, but they're generally at a math level still inaccessible to me: