# what are the properties of gamma function? [closed]

In mathematics, the gamma function (represented by the capital Greek letter $\Gamma$) is an extension of the factorial function, example: $\Gamma(x)$, $\Gamma(ix)$.

What are the physical properties of gamma function (string theories) ?

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## closed as not a real question by Rahul, Norbert, Stefan Hansen, Did, Willie WongJan 30 '13 at 8:13

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This might be a candidate for migration to Physics.SE? – Shaktal Sep 10 '12 at 18:01
What do you mean by the "physical properties" of a function? – Thomas Andrews Sep 10 '12 at 18:02
@Thomas Andrews string is a physics theory!. isn't it?! – alvoera Sep 10 '12 at 18:06
But what does that have to do with the gamma function? The gamma function is not a physical object, so it does not have physical properties. – Thomas Andrews Sep 10 '12 at 18:13
" The gamma function is not a physical object, so it does not have physical properties. " This should be the ultimate answer! – Integral Sep 10 '12 at 18:42

"Physical" properties? As in properties relevant to physics? If that's what the question is about, I might be more explicit about that fact, and then maybe consider migration of physics.SE.

"The properties of the gamma function" is an extensive enough topic for a long and heavy book. I'll mention only one here, and then maybe add more later if I feel like it. $$\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx.$$ Sometimes people ask: Why $\alpha-1$ instead of $\alpha$? Here's one answer; there are probably others. Consider the probability density function $$f_\alpha(x)=\begin{cases} \frac{x^{\alpha-1} e^{-x}}{\Gamma(\alpha)} & \text{for }x>0 \\[12pt] 0 & \text{for }x<0 \end{cases}$$ The use of $\alpha-1$ instead of $\alpha$ makes the family $\{f_\alpha : \alpha > 0\}$ a "convolution semigroup": $$f_\alpha * f_\beta = f_{\alpha+\beta}\tag{1}$$ where the asterisk represents convolution.

Later note: If $X$ and $Y$ are independent random variables with respective densities $f_\alpha$ and $f_\beta$, then $f_\alpha*f_\beta$ is the density of the random variable $X+Y$. So $(1)$ explains why the "shape parameters" of gamma densities simply add up the way they do.

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"The properties of the gamma function" is an extensive enough topic for a long and heavy book. But, please, start with Artin's beautiful, short, skinny one if you're going to read one. :) Also, I strongly suspect Legendre was not thinking about gamma random variables when choosing his notation, though your remark is interesting. :) – cardinal Sep 11 '12 at 3:04

I don't about "physical properties", but the Bohr–Mollerup theorem characterizes the gamma function with simple properties:

$\Gamma(x)$ is the only function $f: (0,+\infty) \to (0,+\infty)$ that satisfies $f(x+1)=xf(x)$ with $\log(f(x))$ convex and $f(1)=1$.

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