Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [gamma-function]

Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions.

0
votes
1answer
32 views

Is this analytic function linked to Barnes multiple gamma function ? Is it entire?

I come across this series of following analytic functions $$ z\mapsto\frac{1}{\Big(\Pi_{k=0}^{n-1}\Gamma(1+ e^{2i\frac{k}{n}\pi}z)\Big)^{\frac{1}{n}}} $$ One can get easily a local (around zero) ...
1
vote
2answers
37 views

Factorials about decimals

How do you get the factorial of a decimal number using a pen and paper if it is possible? Example: Find the factorial of $0.5!$
0
votes
1answer
43 views

Evaluating the integral $\int_0^\infty \mathrm{d}k e^{-i( t - x)k}k^{-i\omega/a}$

I am attempting to compute the vacuum expectation value for the energy density of a particular system (Quantum Field Theory). I come across the following integral $$\int_0^\infty \mathrm{d}k \: e^{-...
-2
votes
0answers
61 views

How can I evaluate $\int_{0}^{\frac{\pi}{2}}\log{\Gamma( \frac{\sin x}{x^2+\pi^2})}$ in closed form? [on hold]

I have tried to evaluate $$\int\limits_0^{\frac{\pi}{2}}\log \left({\Gamma \left( \frac{\sin x}{x^2+\pi^2} \right)} \right)\ dx$$ in closed form , [Invesre symbol calculator] but it didn't ...
2
votes
3answers
41 views

Intermediary steps for this integral of a negative exponential function of arbitrary power

The following result is obtained through Mathematica: $$\int_0^\infty \exp\left(-C\left(\frac{s}{\theta}\right)^\lambda\right)\,\mathrm{d}s = C^{-\lambda^{-1}}\theta\Gamma(1 + \lambda^{-1})$$ where $\...
1
vote
0answers
24 views

Reason for choosing a particular analytical continuation of the factorial

From this answer I know the choice of continuous extention $\ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt\ $ is not unqiue. But is that particular extension the unique best choice in some sense? E.g....
2
votes
1answer
75 views

Why do we evaluate $-x^{z-1}e^{-x}$ as zero when explaining the gamma function through integration by parts?

The gamma function is the integral of $x^{z-1}e^{-x}$ If you integrate by parts you get two terms. The first one is $-x^{z-1}e^{-x}$ and this is bound by infinity and zero. If you plug in infinity, ...
0
votes
0answers
35 views

Integration for special sine-function [duplicate]

I am having trouble integrating the following: $\int_{0}^{\infty} \sin(t)\cdot t^{x-1} ~dt$ for $0 < x < 1$. Does anybody know how that can be done. Ive tried to set it up with powerseries for ...
1
vote
0answers
26 views

Proving that the Gamma function infinite product definition extends its integral form definition

The integral form definition of the Gamma function is as follows. It is valid for all complex numbers with $\mathrm{Re}(z)>0$: $$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x} dx$$ See this Wikipedia page ...
0
votes
1answer
23 views

Integration of digamma function

I was trying to perform the contour integral of the digamma function $\oint\limits_C \psi(z)\,dz$ on the neighborhood (a small circle $-k+re^{it}$, $k \in \mathbb{Z}$ ) of $k$, before actually ...
0
votes
0answers
30 views

How can I demonstrate this series are the same? [duplicate]

How can I know that they are equivalent? $$\sum_{n\in \mathbb Z} e^{-n^2\pi x} = \frac{1}{\sqrt{x}}\sum_{n\in \mathbb Z}e^{\frac{-n^2\pi}{x}} $$ I find this in wikipedia, in an article about the ...
1
vote
0answers
47 views

Generalisation of $\Gamma$ function?

Let the incomplete $\Gamma$-function be defined as: \begin{equation} \Gamma(c+1,x)=\int^\infty_x t^ce^{-t}dt \end{equation} Is there a well studied generalisation of the incomplete gamma function that ...
0
votes
1answer
24 views

Integrating factor for differential equation gamma function

Given the following differential equation \begin{equation} \left(\theta + \frac{1}{1-z}\right)g(z) -\frac{dg}{dz} = \frac{z^\theta}{1-z} \end{equation} It is known that $(1-z)e^{-\theta z}$ is an ...
2
votes
2answers
56 views

Laplace Transform of the incomplete Gamma Function

While looking through this ($178$,$(30)$) Table of Integral Transforms I have come across the Laplace Transform of the Incomplete Gamma Function which is given by $$\mathcal{L}\{\Gamma(\nu,at\}(p)~...
1
vote
2answers
55 views

Find $\alpha(t,a)$, integrable w.r.t. to t, that bounds $\vert e^{-t}t^{a} \frac{t^h - 1}{h} \vert$ ; $a,t > 0$, for all $\vert h \vert < h_0$.

I think I have found an answer to this which I have given at the end but is there a better answer? I am trying to prove differentiability of gamma function $\Gamma(z)$ for $Re(z) > 1$ using ...
3
votes
1answer
85 views

Is there another way of evaluating $\lim_{x \to 0} \Gamma(x)(\gamma+\psi(1+x))=\frac{\pi^2}{6}$

I was messing around with the Zeta-Function and I got what I thought was an interesting limit: $$\lim_{x \to 0} \Gamma(x)(\gamma+\psi(1+x)) = \frac{\pi^2}{6} $$ Where $\Gamma$ is the gamma function, $...
0
votes
0answers
18 views

Bounding ratio of complete and incomplete Gamma functions

I have encountered the following problem and was wondering if anyone had some good hints on it. Given two positive valued sequences $(c_n)_n \sim \Theta(1/n)$ and $(\xi_n)_n \sim \Theta(1)$ -- in ...
0
votes
1answer
47 views

Logarithmic Sum

Is there a closed form for the following sum? $$\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}(-1)^{m+n}\frac{\ln(m+n)}{(m+n)}$$ According to https://www.mathmash.org/contestprob.php?prob=227 it has a closed ...
0
votes
3answers
72 views

Why $\lim_{n\to\infty} \frac{\Gamma\left(n - \frac{1}{2}\right)}{\Gamma\left(n\right)} = e^{-\frac{1}{2}}$

Why is it true that $$\lim_{n\to\infty} \frac{\Gamma\left(n - \frac{1}{2}\right)}{\Gamma\left(n\right)} = e^{-\frac{1}{2}}$$ I only know the integral definition of gamma function. My notes writes $$\...
5
votes
1answer
56 views

What is the name of the function $D(a,x) = \frac{x^a e^{-x}}{\Gamma(a+1)}$?

Does the function $\dfrac{x^a e^{-x}}{\Gamma(a+1)}$ have its own specific name? Temme [1] introduced the function in (3.1) $$D(a,x) = \frac{x^a e^{-x}}{\Gamma(a+1)}.$$ It is the dominant part in many ...
0
votes
1answer
47 views

Asymptotic of gamma function

I came across a quetion: Let $h$ go to zero. What is the asymptotic of $\Gamma(x+o_{p}(h))$ where $x\in(0,2)$? The difficulty is the limitation of x goes to zero. Can I obtain $$\Gamma(x+o_{p}(h))\...
0
votes
2answers
38 views

Trying to Understand $E[X^2]$ for Gamma Distribution

I am trying to understand the following for the gamma distribution: $$E[X^2] = \frac{ \alpha(\alpha+1)}{\lambda^2}$$ I've been looking at the reasoning for $E[X]$ to make sense of what could be ...
0
votes
1answer
58 views

Showing that $\Gamma(x) \Gamma(y) = \Gamma(x+y)\int\limits_0^1 \lambda^{x-1}(1-\lambda)^{y-1}d\lambda$

On page 56 of Titchmarsh's Theory of Functions, Titchmarsh makes the following claim: \begin{align} \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} &= \phi(x,y); (x>0,y>0) \end{align} where \begin{...
4
votes
4answers
141 views

How to derive relationship between Dedekind's $\eta$ function and $\Gamma(\frac{1}{4})$

I am trying to determine in what way to approach finding a connection between Dedekind's Eta Function, defined as $$\eta(\tau)=q^\frac{1}{24}\prod_{n=1}^\infty(1-q^n)$$ where $q=e^{2\pi i \tau}$ is ...
2
votes
1answer
49 views

Show that $\lim_{t \to \infty} \sup_{x \ge 0} \frac{1}{x^2+1} e^{-x} \sum_{n>t} \frac{n x^{n}}{n!}=0$

I am interested in the following limit \begin{align} \lim_{t \to \infty} \sup_{x \ge 0} \frac{1}{x^2+1} e^{-x} \sum_{n\ge t} \frac{ n x^{n}}{n!}=0 \end{align} Note that $\sum_{n>\ge} \frac{ n x^{...
1
vote
0answers
70 views

Yet another bizarre identity involving hypergoemetric functions and gamma functions.

Let $d=4$, $T\ge d$ and $p\ge 0$ be integers. By solving Spectral densities of finite dimensional sample covariance matrices we stumbled on a following identity. \begin{eqnarray} &&\sum\...
3
votes
0answers
72 views

Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular.

Let $A$ be an $n\times n$ matrix whose entries are \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where $0 < \lambda_{1} < \ldots < \lambda_{n}$ and $0 < \mu_{1} < \...
0
votes
1answer
52 views

An integral relation of gamma function

Let, $s=\sigma+it$. For $\sigma >0$ prove that $$\int_0^{\infty}x^{\frac s2-1}e^{-n^2\pi x}\,dx=\frac{\Gamma(s/2)}{n^s\pi^{s/2}}$$ How can I prove this ? I think it can be prove by complex ...
1
vote
1answer
61 views

change of variables with power in integral

I want to compute this integral $$I=\sum_{k=0}^{+\infty} H^k \int_0^t \frac{ u^{(k+1)\alpha-1}}{\Gamma(1-\alpha)\Gamma((k+1)\alpha)}(t-u)^{-\alpha}du$$ With $\alpha\in(0,1)$, $H\in\mathbb{R}^{m\times ...
1
vote
1answer
40 views

How to solve recurrence equation by gamma function?

I have a recurrence equation and I have no idea how to solve it. $$a(n)=(1-\frac{1}{n}+\frac{X}{n})a(n-1)+\frac{2}{n}$$ $$a(1)=1$$ where X is a constant I tried to put it into Wolfram and I got the ...
0
votes
4answers
82 views

The sum of logarithmic series

I will be very grateful for help and suggestions how to calculate the sum $$\sum\limits_{n=2}^{\infty}\frac{\log(n)}{n(n-1)}$$
1
vote
1answer
34 views

How to transform the integration in Gamma function from 0-1 to 0-infinity or vise versa

Here's something basic on the transformation of Gamma function and I'm wondering if someone could explain to me how the following works: we know the function has two formulations: $\Gamma(x) = \int_{...
2
votes
3answers
287 views

Quick evaluation of the Gamma function?

I am given an exercise about the beta distribution, with a solution: EXAMPLE 4.11 A gasoline wholesale distributor has bulk storage tanks that hold fixed supplies and are filled every Monday. Of ...
3
votes
0answers
77 views

Why does $\int_0^1 \frac{\ln(\ln(p))}{1+p^2}dp$ Converge?

I was messing around with the Dirichlet Beta Function and was able to get a formula: $$\int_0^1 \frac{\ln(\ln(p))\ln(p)^{x-1}}{1+p^2}dp = \Gamma(x)(-1)^x(\beta'(x)-\beta(x)(i\pi +\psi(x)) $$ where $\...
7
votes
3answers
221 views

Show that $2^{ax}\frac{\Gamma((a+1)x)}{\Gamma(x)}$ is an increasing function

I would like to show that the following function \begin{align} f_a(x)=2^{ax}\frac{\Gamma((a+1)x)}{\Gamma(x)} \end{align} is an increasing function in $x$ for $x \ge 0$ for any fixed $a>0$. I did ...
0
votes
0answers
31 views

How can I reduce the definition of the multivariate gamma function?

This question is motivated by the discussion arising from a previous question. It is intended to provide a more concrete example highlighting the some of the rationale for asking that more general ...
1
vote
2answers
51 views

what is the value of i factorial using the complex number system? [duplicate]

what is the value of i factorial? "I" belongs to the complex number system. Thanks for helping me out with this problem.
0
votes
1answer
33 views

Gamma function results

The gamma function is known to follow the below recurrence condition for all positive real numbers- $$\Gamma\big(x+1\big)=x~\Gamma\big(x\big).~$$ Consider the function $\Gamma\big(n+\frac{1}{2}\big)$...
1
vote
1answer
91 views

An integral identity involving Gamma function [closed]

I started with the integral: $$\tag{1} \int_{-1}^1 \frac{\left(1-u^2\right)^\frac{D-4}{2}\; du}{1+A u} $$ I evaluated the integral in Mathematica: ...
2
votes
1answer
34 views

Analytic Continuation of the incomplete gamma function

I have the following expression for $\alpha,z>0$: \begin{equation} \pi\mathrm i-\Gamma(\alpha)(-1)^\alpha\Gamma(1-\alpha,-z). \end{equation} In the context of the problem I am looking at this ...
0
votes
0answers
19 views

Generalization of hypergeometric type differential equation

I am aware that hypergeometric type differential equations of the type: can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, ...
3
votes
0answers
49 views

Closed form for infinite series $\sum_{n=1}^\infty \prod_{j=1}^n \left[1-(\tfrac{j}{n}u+v)^2\right]^{-1} x^n$

I've been struggling to find a closed form for the following series that depends on $u$ and $v$ as parameters: $$ f(u,v,x) = \sum_{n=1}^\infty c_n x^n $$ with $$ c_n = \frac{1}{\prod_{j=1}^n \left[...
1
vote
1answer
28 views

Show that $B(\alpha,\alpha)=2\int_0^{\frac{1}{2}}(\frac{1}{4}-(\frac{1}{2}-x)^{2})^{\alpha-1}dx$

Show that $B(\alpha,\alpha)=2\int_0^{1/2}(\frac{1}{4}-(\frac{1}{2}-x)^{2})^{\alpha-1}dx$ Where $B(\alpha,\beta)=\int_0^{1}x^{\alpha-1}(1-x)^{\beta-1}dx$ I tried in many ways $$B(\alpha,\alpha) =\...
4
votes
1answer
79 views

Evaluating $\int_0^\infty \sin (x^p) \, dx$ via gamma function

I was evaluating a this complex integral via gamma function: $\int_0^\infty \sin (x^p) \,dx$ $\;$for $p \gt 1$, so I expressed it as an imaginary part of $\int_0^\infty \exp(-ix^p) \, dx$ $\;$for $p \...
3
votes
0answers
79 views

Proof that Integral over the arc vanishes as $R\rightarrow\infty$ in Inverse Mellin Transform of $\Gamma(s)$

It´s a very well know result that $$e^{-x}=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \Gamma(s)x^{-s}ds$$ In order to solve this integral we have to close the contour to the left and show ...
1
vote
0answers
25 views

Approximating Log(Gamma(z)) for small z as Log(Gamma(z + 1)) - Log(z)

I'd like to implement a numerical approximation to the log Gamma function, and I found Gergő Nemes' approximation described here: https://en.wikipedia.org/wiki/Stirling%27s_approximation. This seems ...
1
vote
1answer
50 views

Manipulation of gamma functions

Wolfram Alpha tells me for instance $$ \frac{\Gamma(6-1/4)}{\Gamma(12+5/4)}-\frac{\Gamma(12-1/4)}{\Gamma(9+5/4)}=\frac{133259008 \sqrt{2} \pi}{1020857565\Gamma(1/4)^2}. $$ I am now looking for a ...
2
votes
1answer
79 views

Gamma Identities from Inverse Transform

I have been working on a transform (the notes are a little rough and the hosting website has some fraction formatting issues recently) but the idea is there: Basically, the transform $\mathcal{I}_x[f(...
3
votes
1answer
37 views

How to derive the domain of Gamma function

I was reading about the Gamma function, however, I have some trouble to figure out why the domain of $$\Gamma(\alpha)= \int_0^{\infty} x^{\alpha -1}e^{-x}\,dx$$ How can I get $ \alpha > -1$?
4
votes
1answer
56 views

Showing that $\Gamma(x)\Gamma(y) = \Gamma(x+y)\beta(x,y)$ via change of variable

From $$\Gamma(x)\Gamma(y)=\int_0^\infty e^{-t}t^{x-1} \left( \int_0^\infty e^{-t} s^{y-1} ds \right) dt,$$ use a change of variable $s=ut$ to show $$\Gamma(x)\Gamma(y)=\Gamma(x+y)\beta(x,y).$$ Let ...