Gamma function for $0 < n < 1$

Question

The gamma function is defined by : $\Gamma(n) = (n-1)!$ and it says this function is defined on : $\mathbb{R}^{+}$ whereas this definition of the gamma function is of course not defined when $0 < n <1$ ?

Answer 1

The gamma function only satisfies $\Gamma(n)=(n-1)!$ for positive integers.

For non-integer numbers you must go back to the actual definition of the gamma function, which is $$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$ If you evaluate this for $x\in\Bbb Z^+$, you will get the previous result with $(n-1)!$, but for non-integer positive (or negative) real numbers you must use this integral.