TITLE: Why isn't the gamma function defined so that $\Gamma(n) = n! $?
QUESTION [35 upvotes]: As a physics student, I have occasionally run across the gamma function
$$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$
when we want to generalize the concept of a factorial.  Why not define the gamma function so that 
$$\Gamma(n) = n!$$
instead?
I realize either definition is equally good, but if someone were going to ask me to choose one, I would choose the second option.  Are there some areas of mathematics where the accepted definition looks more natural?  Are there some formulas that work out more cleanly with the accepted definition?

REPLY [7 votes]: $$
\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx.
$$
Why $\alpha-1$ instead of $\alpha$?  Here's one answer; there are probably others.  Consider the probability density function
$$
f_\alpha(x)=\begin{cases} \dfrac{x^{\alpha-1} e^{-x}}{\Gamma(\alpha)} & \text{for }x>0 \\[12pt] 0 & \text{for }x<0 \end{cases}
$$
The use of $\alpha-1$ instead of $\alpha$ makes the family $\{f_\alpha : \alpha > 0\}$ a "convolution semigroup":
$$
f_\alpha * f_\beta = f_{\alpha+\beta}
$$
where the asterisk represents convolution.