How to come up with the gamma function? It always puzzles me, how the Gamma function's inventor came up with its definition
$$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$
Is there a nice derivation of this generalization of the factorial?
 A: $$
\int e^{ax} dx = \frac{1}{a} e^{ax} + c 
$$
Take $\left .\frac{d}{da}\right |_{a=1}$ on both sides $n$ times, and algebra to get rid of $(-1)^n$, you'll have an integral equal to $n!$.
This is an intuitive way to get the Gamma function.  You've shown that for integers it holds from this simple derivation.  
Mathematicians then went through a great deal of work to show that it holds true for allot more than just the integer case.
A: I guess you can say this is yet another application of the power of integration by parts (and I am guessing that is how the integral formula "was come up with" initially).
If you are trying to find the antiderivative of $P(t) e^t$, where $P(t)$ is a polynomial, integration by parts arises naturally and I would say it(integral of $P(t) e^t$) is quite natural to encounter during ones study of mathematics. And if you actually work it out, you notice the factorial like recursion. We can rid of the "non-integral" parts of the integration by parts formula by using the limits $0$ and $\infty$.
If $I_n = \int_{0}^{\infty} t^n e^{-t} \text{dt}$ then integration by parts gives us
$$I_n = -e^{-t}t^n|_0^{\infty} + n\int_{0}^{\infty}  t^{n-1} e^{-t} = nI_{n-1}$$
so if 
$f(x) = \int_{0}^{\infty} t^x e^{-t} \text{dt}, \quad x \ge 0$
then
$f(x) = x f(x-1), \quad x \ge 1$.
Also, we have that $f(0) = 1$, thus the integral definition agrees with the factorial function at the non-negative integers and can serve as a real extension for factorial.
Using Analytic continuation its domain can be extended further.
A: Here is a nice paper of Detlef Gronau Why is the gamma function 
so as it is?.
Concerning alternative possible definitions see Is the Gamma function mis-defined? providing another resume of the story Interpolating the natural factorial n! .
Concerning Euler's work Ed Sandifer's articles 'How Euler did it' are of value too, in this case 'Gamma the function'.
A: In Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz by Philip J. Davis in The American Mathematical Monthly , Dec., 1959: Apparently, Euler, experimenting
with infinite products of numbers, chanced to notice that if n is a positive integer,
$$\small n!=\Bigg [ \left(\frac 2 1 \right)^n \frac1{n+1}\Bigg ]\,\Bigg [ \left(\frac 3 2 \right)^n \frac2{n+2}\Bigg ]\,\Bigg [ \left(\frac 4 3 \right)^n \frac3{n+3}\Bigg ]\cdots$$
and succeeding in transforming this infinity product into an integral, extending the factorial beyond integers, upon noticing that for certain values the infinite product yielded $\pi, $ suggesting areas of a circle.
But there is a really neat intuition already expressed in one of the answers, and beautifully presented by Robert Andrew Martin here, simply expanding the integral part in the integration by parts of a polynomial modulated by an exponential. In essence, the counterpart of the factorials in Taylor series.
For instance for $x^4$ (leaving constant of integration out):
$$\begin{align}\small
\int x^4\; e^{-x} \; dx &\small= -x^4\;e^{-x} +\int 4\,x^3\;e^{-x}\;dx\\
&\small = -x^4 \; e^{-x}-4\,x^3\;e^{-x} +\int_0^\infty 4\cdot 3\,x^2\;e^{-x}\;dx\\
&\small = -x^4 \; e^{-x} -4\,x^3\;e^{-x}  -4\cdot 3\,x^2\;e^{-x} +\int 4\cdot 3\cdot 2\,x\;e^{-x}\;dx\\
&\small =-x^4 \; e^{-x} -4\,x^3\;e^{-x}  -4\cdot 3\,x^2\;e^{-x}  -4\cdot 3\cdot2\,x\;e^{-x} - \underbrace{4\cdot 3\cdot 2\cdot 1}_{4!}\;e^{-x}
\end{align}$$
Generalizing and integrating from $0$ to $\infty:$
$$\small\int_0^\infty x^n\; e^{-x} \; dx=-x^n \; e^{-x} -n\,x^{n-1}\;e^{-x}  \cdots - \underbrace{n\cdot (n-1)\cdots 3\cdot 2\cdot 1}_{n!}\;e^{-x}\;\;\Bigr|_{x=0}^\infty=n!$$
which can immediately be extended beyond integers as $\displaystyle \small x! = \int_0^\infty t^x\; e^{-t} \; dt$
essentially the gamma function, except for the accepted slightly different definition: $$\Gamma(x)=\int_0^\infty t^{x-1}\; e^{-t} \; dt$$
that makes $\small (x-1)!=\Gamma(x).$
