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) ?
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) ?
"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.
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 $f(1)=1$.