The Gamma function and the Pi function

Solution 1:

Since you mention Laplace transforms, in its current form $\Gamma(s)$ is the Mellin transform of $e^{-x}$.

Here is another reason which is perhaps the most convincing. The Haar measure of a subset $S\subset \mathbb{R}^\times$ of the multiplicative group of real numbers is $\int_{x\in S} \frac{dt}{t}$, so the measure $\frac{dt}{t}$ over the real line is natural. The Gamma function is an analogue of a Gauss sum, and is the integral of multiplicative function $x^s$ against the additive function $e^{-x}$ over the measure of the group.

This problem was posed on Math Overflow, and received a large number of upvotes there. Take a look at the answers appearing on this thread: https://mathoverflow.net/questions/20960/why-is-the-gamma-function-shifted-from-the-factorial-by-1

Solution 2:

This subject is discussed in the book The Gamma Function by James Bonnar. The $\Pi(z)=z!$ notation is due to Gauss and is sometimes encountered in older literature. The notation $\Gamma(z+1)=z!$ is due to Legendre. Legendre's motivation for the normalization does not appear to be known. Cornelius Lanczos called it "devoid of any rationality" and would instead use $z!$. Legendre's formula does simplify a few formulas, but complicates most others.