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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?

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    The link below provides the history of how the Gamma function was invented. It took many years before getting into the form that we know it now. https://drive.google.com/open?id=0B4W-gdhbNpsDaTNvbF9VcGgyR1E2016-05-28

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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'.

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    @DanielG: Thanks for the update! Concerning the last paragraph we may use ['How Euler did it'](http://eulerarchive.maa.org/hedi/index.html) and ['Gamma the function'](http://eulerarchive.maa.org/hedi/HEDI-2007-09.pdf) from the irreplaceable ["The Euler Archive"](http://eulerarchive.maa.org). Cheers,2016-06-23
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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.

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    @FUZxxl: I have added a paragraph. See the edit.2012-03-11
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$ \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.

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    To be more precise, we have\int_0^\infty e^{ax}~\mathrm dx=-\frac1a\quad \forall a<0Differentiate both sides $n$ times to get$\int_0^\infty x^ne^{ax}~\mathrm dx=\frac{(-1)^{n+1}n!}{a^{n+1}}$Now sub in $a=-1$ to get$\int_0^\infty x^ne^{-x}~\mathrm dx=n!$2017-08-20