TITLE: The Gamma function and the Pi function
QUESTION [23 upvotes]: I have been studying differential equation, in particular special functions.
Euler's Gamma function, and Gauss's Pi function are essentially the same, differing only by an offset of one unit.
for  $z\in\mathbb C,\qquad {\frak R} (z)>0$
$$\Gamma(z)=\int\limits_0^\infty x^{z-1}e^{-x}\,\mathrm dx$$
$$\Pi(z)=\Gamma(z+1)=\int\limits_0^\infty x^ze^{-x}\,\mathrm dx$$
Both extend the notion of the factorial (which is only defined for positive integers).
$$\Gamma(z+1)=\Pi(z)=z!,\qquad z\in\mathbb Z \geq0$$

The Pi function appears to be a more natural analog of the factorial (It dosen't introduce the unit offset).
My text book exclusively uses the Gamma function, and dosen't mention the Pi function at all.
I was wondering if there is any good reasons to focus on the Gamma function (presumably it makes some calculations simpler further down the line).

The best reason I can come up with on my own is that for Laplace transforms 
$$\mathcal L\big\{ t^r\big\}=\frac{\Pi(r)}{s^{r+1}}=\frac{\Gamma(r+1)}{s^{r+1}},\qquad r\geq-1 \in\mathbb R$$
Using the Gamma function here preserves some symmetry.
I am not sure it this is the reason or if there are some subtleties I am completely missing.

REPLY [5 votes]: 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.