Questions tagged [gamma-function]

Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions. The Gamma function is a specific way to extend the factorial function to other values using integrals.

Gamma function, generalization of the factorial function to nonintegral values, introduced by the Swiss mathematician Leonhard Euler in the $18^{th}$ century.

Definition: The gamma function, denoted by $\Gamma$, is defined by \begin{equation*} \Gamma(z)=\int^{\infty}_{0}x^{z-1}e^{-x}\ \mathrm dx, \end{equation*} where $z$ is a complex number whose real part is greater than $0$. This integral function is extended by analytic continuation to all complex numbers except the non-positive integer. The reason for $z-1$ instead of $z$ in the exponent is to reflect the fact that $1/x$ is not improperly integrable on either $(0,1]$ or $[1,\infty)$.


$1.~$ For $\Re(z)>0$ the integral is convergent, i.e. $\Gamma$ is well-defined. Also, $\Gamma(z)>0$ for $z>0$.

$2.~$ $\Gamma(z+1) = z \Gamma(z)$ and if $n\in\mathbb{Z}^+$, $\Gamma(n)=(n-1)!$. This allows us to extend the definition to any $z\in\mathbb{C}$, except non-positive integers.

$3.~$ $\Gamma(1)=1$

$4.~$ $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$

$5.~$ $\displaystyle{ \Gamma(z)\Gamma(1-z) = \pi \csc(\pi z)}$

$6.~$ $\log(\Gamma(z))$ is convex

$7.~$ $\Gamma(z)$ is analytic for $s>0$

$8.~$ $\Gamma(z)$ admits a Weierstrass product representation: $$ \Gamma(s) = \frac{e^{-\gamma z}} z \prod_{n=1}^\infty \left(1 + \frac z n \right)^{-1} e^{z/n}, $$where $\gamma$ is the . In particular, $\Gamma(s)\neq 0$ for any complex $z$.

The famous Bohr-Mollerup theorem says that properties $1,3,6$ uniquely characterize $\Gamma$.

Here is a quick look at the graphics for the gamma function along the real axis.

enter image description here


The gamma function shows up in many, seemingly unrelated, fields of mathematics. In particular, the generalization of the factorial provided by the gamma function is helpful in some combinatorics and probability problems. Some probability distributions are defined directly in terms of the gamma function. For example, the gamma distribution is stated in terms of the gamma function. This distribution can be used to model the interval of time between earthquakes. Student's $t$ distribution, which can be used for data where we have an unknown population standard deviation, and the chi-square distribution are also defined in terms of the gamma function.


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Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ , coming from the Gamma function. I have a mathematical…
Qiaochu Yuan
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Why is Euler's Gamma function the "best" extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"? In particular, I'm looking for…
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Why does "Turn! Turn! Turn!" equal 241217.524881?

If you search for "Turn! Turn! Turn!" on Google, then the second result is this YouTube video of The Byrds performing the Pete Seeger song of that name. But the first result is Google's internal calculator displaying "241217.524881". With a bit of…
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Stirling's formula: proof?

Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = \sqrt{2 \pi}$. What is a good way of doing this? Could…
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Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?

It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ \Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\text{ ?} $$ (I'll post my own…
Michael Hardy
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Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \log \Gamma(z) - \log…
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Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?

Is it possible to simplify this expression? $$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$ Is there a systematic way to check ratios of Gamma-functions like this…
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How to find the factorial of a fraction?

From what I know, the factorial function is defined as follows: $$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$ And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction, like, for instance, $\frac{1}{2}!$, which they…
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How do you prove Gautschi's inequality for the gamma function?

A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > 0,\; 0 < s < 1$$ Unfortunately, the paper that the DLMF is…
J. M. ain't a mathematician
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Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$

Is there a closed form for the following infinite product? $$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
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How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$

Define the multifactorial function $$n!^{(k)}=n(n-k)(n-2k)\cdots$$ where the product extends to the least positive integer of $n$ modulo $k$. In this answer, I derived one of several analytic continuations of this function to the real numbers, which…
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a conjectured continued fraction for $\tan\left(\frac{z\pi}{4z+2n}\right)$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$ is…
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Prove that $\frac{1}{\phi}<\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx< \frac{24+\sqrt{2}}{41} $

I'm sure that's a coincidence, but the Laplace transform of $1/\Gamma(x)$ at $s=1$ turns out to be pretty close to the inverse of the Golden ratio: $$F(1)=\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx=0.61985841414477344973$$ Can we prove analytically…
Yuriy S
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Evaluating the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Bigg[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Bigg]- \frac{4}{3} \zeta(3) \, , $$ where $\psi_{1}(x)$ is…
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Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ This question was posted as part of this question:…
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