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It always puzzles me, how the Gamma functions's inventor came up with it's 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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Related: math.stackexchange.com/questions/1537/… –  user17762 Mar 11 '12 at 21:29
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Also related: math.stackexchange.com/questions/3444/… –  cardinal Mar 11 '12 at 22:22

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up vote 5 down vote accepted

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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@FUZxxl: Well it provides the letters from Euler where this generalization appears (Euler usually didn't give names to his functions, Gamma and +1 came later...). –  Raymond Manzoni Mar 11 '12 at 21:38
    
Sorry. I found out that the article actually contains an answer to my question, thus I removed my post. –  FUZxxl Mar 11 '12 at 21:47
    
@FUZxxl: no problem and fine reading! –  Raymond Manzoni Mar 11 '12 at 21:52

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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Thank you for that answer although you don't point out how one get's to $\int_0^\infty t^ne^{-1}\;\mathrm dt$. –  FUZxxl Mar 11 '12 at 21:48
    
@FUZxxl: I don't understand. I interpreted your question as "how did one come up with the integral". Or as you asking for why $\Gamma(x+1) = x!$ and not $\Gamma(x) = x!$? –  Aryabhata Mar 11 '12 at 21:50
    
YOur first interpretion is right. But your answer shows only that the integral satisfies the recurrence relation $\Gamma(x+1)=x\Gamma(x)$ and does not show how one can derive that integral. –  FUZxxl Mar 11 '12 at 21:57
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@FUZxxl: I have added a paragraph. See the edit. –  Aryabhata Mar 11 '12 at 21:58
    
Okay. Thank you! –  FUZxxl Mar 11 '12 at 22:02

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