Computing of the Gamma Function

Question

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:

• Understanding the Gamma function? [duplicate]
• Intuition for the definition of the Gamma function?
• How to come up with the gamma function?
• Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

Answer 1

For real $x$, the gamma function provides a continuous (indeed, even smooth) function that gives the factorials at the positive integers. However, you can choose points in between the positive integers and define your function to be whatever value you want there, and still find a smooth function that fits those values plus gives factorials at the positive integers. So we want some restrictions to define a "good" gamma function. One property of factorials is that $x! = x \cdot (x-1)!$ so it makes sense to insist that $\Gamma (x+1) = x \Gamma (x)$. However, we can still define an essentially arbitrary smooth function between $x=1$ and $x=2$ that takes on factorial values at those end points, and then extend the function to all positive $x$ by using the relation $\Gamma (x+1) = x \Gamma (x)$. So yes, in a sense, the values that the Gamma function assigns to non-integer positive reals are essentially arbitrary. The main reason why the Gamma function is preferred is that it appears so often in other mathematical formulas and studies.