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Questions tagged [gamma-function]

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

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Step in the proof of the continuity of the Gauss Gamma function

Define the Gamma function as \begin{align} \Gamma(z):= \frac{1}{G(z)} \quad \forall \, z\in \mathbb{C}\setminus \{0, -1, -2, ... \} \end{align} where \begin{align} G: \mathbb{C} \rightarrow \mathbb{C}...
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Laurent expansion of the gamma function at negative integers

It is known that the gamma function has the non-positive integers as its simple poles. And we know the residues at these poles. $$ Res(\Gamma(z=-n)) = \frac{(-)^n}{n!} . $$ This means, its Laurent ...
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24 views

Gamma Distribution and Chi Square Distribution

Actally i have 2 question for you guys. It's not a homework. I just curious. What is the difference of Erlang distribution and Gamma distribution. On Wikipedia it's said if Erlang is Gamma ...
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Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$ z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)! $$ ...
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Calculating $\int_{S^{n-1}}x_1^2dS$

Denote $x=(x_1,...,x_n)$. I want to calculate the following integral: $$\int_{S^{n-1}}x_1^2dS$$ Where $S^{n-1}$ is the $n$ dimensional sphere. I was given a hint to use the fact that the volume of $...
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25 views

Question about the Gamma function

My question is fairly simple: I was wondering if $\,\,\,\Gamma\left(\pi\right) = 2.2880377\ldots\,\,\,$ had any special meaning. Is it irrational ?. transcendental ? is it useless ? ...
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41 views

The integrals $\int_0^\infty \frac{\mathrm{d}x}{(a+bx^c)^d(e+fx^g)^h}$ and $\int_0^\infty \frac{x^e}{(a+bx^c)^d}\mathrm{d}x$

I am interested in improper integrals of rational functions. For example, I have found that $$\large{\int_0^\infty \frac{\mathrm{d}x}{(a+bx^c)^d}=\frac{\Gamma(\frac1c+1)\Gamma(d-\frac1c)}{\Gamma(d)a^{...
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Looking for a proof of an interesting combinatorial identity

Trying to generalize a combinatorial identity I have ended up with the following expression: $$ \sum_{\{k\}_L}\binom\nu K K!\prod_l \frac1{k_l!}\binom \mu l^{k_l}=\binom{\nu\mu}L,\tag1 $$ where the ...
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difficulty evaluating solving this gamma function value

I am aiming to solve the following $$\prod_{n=1}^{\infty} \left(1-\frac{1}{(2n)^3} \right) $$ Note its similarity to $$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^3} \right)=\frac{\cosh(\frac{\pi}{2}\...
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Integral involving upper incomplete gamma function, exponential and rational funcitons

Related to these questions here and here, I found a different form of the integrals, which result in $$f(x)=\frac{(-1)^n \,2^n}{\pi\,\lambda^{n+1}} \,\mathrm{e}^{-\lambda\,x}\int_{-\infty}^{\infty}\...
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How can i evaluate the following Gamma Function value

Over the last few days i have been studying equation (20) in the following (http://mathworld.wolfram.com/InfiniteProduct.html) A special case of this formula is given as follows $$ \prod_{n=2}^{\...
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Expectation of function Non-central Chi-square RV

I am trying to calculate the expected value of the function $$f(X) = \gamma \bigl( a,\frac{b}{cX+d} \bigr)~,$$ w.r.t random variable $X$, where $X$ has non-central Chi-square distribution with a ...
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1answer
58 views

$\int_{-1}^1 (t-1)\left(e^\frac{1}{\Gamma(t)}-1\right)dt$

I am looking for a way to evaluate the integral $$ \int_{-1}^{1}\left(t - 1\right)\left[\mathrm{e}^{1/\Gamma\left(t\right)} - 1\right]\mathrm{d}t $$ This integral appears to almost have a sort of ...
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Compute $ \xi_p= \prod_{n=1}^{\infty} (1+\frac {1}{n^p})$

The main question I want to ask is inspired from this question Find the value of $$\prod_{n=1}^{\infty} \left(1+\frac {1}{n^2}\right)$$ Now, I have solved this question easily using product ...
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1answer
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Algebra. Solving for a given gamma function: $\ln L = n \ln(\Gamma(a+1)) - n \ln (\Gamma(a)) + (a-1) \sum_{i=1}^{n} \ln x_i$

$\ln L = n \ln(\Gamma(a+1)) - n \ln (\Gamma(a)) + (a-1) \sum_{i=1}^{n} \ln x_i$ so given this I want to solve the derivative for $a$ then solve for $a$, $\ln L = 0$ $0 = \frac{n(\Gamma(a+1)')}{\...
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82 views

Integral of $ \int x^{n-1}W(x)dx $

How to prove that : $$ \int x^{n-1}W(x)dx = \frac {x^ne^{[-nW(x)]}[-nW(-x)]^{-n}[n\Gamma(n+1, -nW(x)- \Gamma(n+2, -nW(x))]} {n^2} $$ Where $W(x)$ is the Lambert-W function https://en.wikipedia.org/...
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1answer
20 views

Asymptotics of $\Gamma(z+\alpha)/\Gamma(z+\beta)$ when all three parameters $\to \infty$

Lots of results are known about the asymptotic ratio of two gamma functions when $z\to\infty$ and $\alpha,\beta$ are constants, the most basic one being: $$\frac{\Gamma(z+\alpha)}{\Gamma(z+\beta)} \...
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Integral Representation of different Gamma-Functions

I came across the relation $(Γ(x) Γ(y))/Γ(x + y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$ Can someone tell me how to prove this? Thanks!
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$2 \cdot \int_0^1 \log\big(\Gamma(x)\big) \cdot \sin(2 \pi n x) dx = \frac{\gamma + \log(2 \pi) + \log(n)}{n \pi}$

I want to proof the series of kummer. Therefore I want to show $$2 \cdot \int_0^1 \log\big(\Gamma(x)\big) \cdot \sin(2 \pi n x) dx = \frac{\gamma + \log(2 \pi) + \log(n)}{n \pi},$$ where $\gamma$ is ...
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1answer
29 views

Ratio of Gamma function equality.

Let $m,n\in\mathbb N$ with $m<n$ and $0<s<1$, $1\leq p<\infty$. Is the following holds $$\frac{\Gamma(\frac{sp+p+n-2}{2})\Gamma(\frac{n-m+sp}{2})}{\Gamma(\frac{n+sp}{2})\Gamma(\frac{sp+p+n-...
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1answer
46 views

closed form of the following integral :$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x) dx$?

I have tried to evaluate this:$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x)$ using the the following formula $$2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\Big( \frac{...
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33 views

Taylor series error?

I was playing around with Taylor series and a shifted Gamma Function and found something that doesn't work, and I'm not sure what I'm doing wrong. Is it some uniform continuity problem? Let: $$\Pi(n)...
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70 views

inequality for ratio of Gamma functions

Let $N,q$ be natural numbers. Find the best upper and lower bound (non-asymptotic) for $$ \frac{\Gamma(2q)}{\Gamma(q)}\frac{\Gamma(q+N/2)}{\Gamma(2(q+N/2))}=\frac{\prod_{k=1}^q(2q-(2k+1))}{\prod_{\...
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1answer
29 views

Why is the Gamma function defined by the definite integral for $\Re(z) >0$?

Gamma Function gives the integral definition of the gamma function as \begin{equation*} \tag{1} \Gamma(z) := \int_{0}^\infty e^{-t}t^{z-1}\,\,dt, \text{ for } \Re(z) > 0. \end{equation*} Why ...
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1answer
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Is the Gamma function defined by $\int_{0}^\infty e^{-t}t^{z-1}\,\,dt?$

Should you write \begin{equation} \Gamma(z) := \int_{0}^\infty e^{-t}t^{z-1}\,\,dt, \text{ for } \Re(z) > 0, \tag{1} \end{equation} or \begin{equation} \Gamma(z) = \int_{0}^\infty e^{-t}t^{z-1}\,\...
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2answers
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What is $\int_0^\infty \log\left(\Gamma(x)\right)e^{-sx}dx$?

I have been interested in the Laplace Transform of the log-Gamma function$$\int_0^\infty \log\left(\Gamma(x)\right)e^{-sx}dx$$ By expanding the Gamma Function into its Weierstrass Form we get \begin{...
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Digamma function question

I just learned online about Polygamma functions, and I want to know what $x$ equals (and how to get it) when $\psi(x)=1$ and $x>1$.
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Application of Legendre's duplication formula

I am reading the book "Special functions: an introduction to classical functions of mathematical physics" by Nico M. Temme and I'm having trouble understanding how to find a constant (in page 62). I ...
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Solving $1=\Gamma\left(\frac{d}{2}\right)-\Gamma\left(\frac{d}{2}, 2t \right)$

Is it possible to solve the following equation that contain the incomplete Gamma function? $$1=\Gamma\left(\dfrac{d}{2}\right)-\Gamma\left(\dfrac{d}{2}, 2t \right)$$ I would like to find the value ...
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37 views

Simplify $\lim_{n \to \infty}\frac{(1-A)(1-B)}{(2-A-B)(A-B)}=\frac{\gamma(\gamma-1)}{2\gamma-1}?$

Let: $$A=\Gamma\left(1+\frac{1}{\Gamma\left(\frac{1}{n}\right)}\right)$$ $$B=\Gamma\left(1+\frac{1}{1+\frac{1}{\Gamma\left(\frac{1}{n}\right)}}\right)$$ I spent a few days trying to work on the $\...
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0answers
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At which value (over $\mathbb{R}^+$) is the gamma function strictly increasing? [duplicate]

The title basically states the whole question^^ Over the positive reals, at which point does the gamma funtion actually start increasing? I tried using the difference of $\Gamma_n-\Gamma_{n-1}$ and ...
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3answers
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Prove that $\int_0^\infty e^{-x} \ln x d x = - \gamma $ [duplicate]

I can see it is right by using some knowledge of the Gamma function. We have $$ \Gamma(\alpha ) = \int_0^\infty e^{-x} x^{\alpha - 1 } dx . $$ Differentiating with respect to $\alpha$, we get $$ \...
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1answer
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Proving that $\Gamma(\frac{1}{2})=\sqrt(\pi)$ using the expected value of standard normal variable (integral calculation)

I'm looking to prove that $\gamma$$(\frac{1}{2})=\sqrt(\pi)$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt(2\pi)}e^{\frac{-z^2}{2}} z^2 dz$ (where $Z$ is a standard normal ...
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Similarity between $e^x$ power series and Gamma function integral?

The power series for $e^x$ is as follows. $$e^{x} =\sum ^{\infty }_{n=0}\frac{x^{n}}{n!}$$ If we define $n! = \Gamma(n+1)$, then we have $$n!=\int ^{\infty }_{0} x^{n} e^{-x} dx.$$ An extremely ...
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Asymptotic for the gamma function on vertical lines

On page 135 of Joerg Bruedern's "Einfuehrung in die analytische Zahlentheorie" he claims that Stirling's formula implies for fixed $\sigma <0$, any $t\geq 1$, and some constant $C$ (I assume ...
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An equation with Gamma Euler function in critical strip

Let $$ D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \} $$ that is the critical strip without critical line. I have to find if the following equation, with ...
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1answer
89 views

Integral of $x^2 \sin(x^2)$

I was playing around learning the SymPy syntax (python library) e then I saw an example that intrigued me. $$\int x^2\sin(x^2)dx = \frac{5x\cos(x^2)\Gamma(\frac{5}{2})}{8\Gamma({\frac{9}{4})}} + \...
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2answers
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Evaluate the limit without using Stirling Formula

I want to prove that $$\lim_{n \to \infty} p \frac{n^{(p+1)/2} (n!)^p p^{np+1}}{(np+p)!} = p^{1/2} (2\pi)^{(p-1)/2}$$ So I am working the book The Gamma Function by Emile Artin. In the book this ...
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Graphing Sum of Taylor Series using Gamma Function

For a school project I have set out to create a function which shows how the difference in the area underneath the curve of the two graphs of $f(x)=e^x$ and $f(x)=\sum_{n=0}^k \frac{x^n}{n!}$, changes ...
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2answers
73 views

Evaluation of special exponential integrals

I am trying to prove that $$\int_{\mathbb{R}} x e^{-x} \cdot e^{-e^{-x}-x} = 1 - \gamma$$ and $$\int_{\mathbb{R}} x^2 e^{-x} \cdot e^{-e^{-x}-x} = \pi^2/6 - 2\gamma + \gamma^2$$ I have tried to ...
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1answer
113 views

Integral involving incomplete beta function

I have the following integral, $$\int_{0}^1x^{a-1}(1-x)^{b-1}B_x(c,d)dx$$ where $B_x(c,d) = \int_{0}^xt^{c-1}(1-t)^{d-1}dt$ is the incomplete beta function, and $a,b,c,d>0$. Question: Does this ...
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1answer
19 views

Dimensional regularization and expansion of gamma function

In my calculations, I used dimensional regularization, i.e. replace $d\rightarrow d-\epsilon$ and calculated the divergent integral. Then, I would like to expand the answer into seriers by $\epsilon$ ...
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How can you define the exponential and logarithmic integrals in terms of the incomplete gamma function?

https://www.wolframalpha.com/input/?i=gamma(0,x) https://en.wikipedia.org/wiki/Exponential_integral https://en.wikipedia.org/wiki/Logarithmic_integral_function For the Euler integral definition, it ...
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19 views

Upper bound for the complex Beta function

Is there any work or reference regarding upper bounds for the complex beta function defined by \begin{equation} B(x,y)=\frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}, \end{equation} for $\Re{x} >0$ and $...
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1answer
19 views

Need Help with Some Advanced Integration By Parts Methods

Note: I am asking this question for someone to check my work for me. The problem started out with me finding z! which is equal to the $\mathbf P \mathbf i$ $\mathbf f \mathbf u \mathbf n \mathbf c \...
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40 views

Derivative of $\Gamma (1)=-\gamma$

How to compute $\Gamma'(1)=-\gamma$ where $\gamma$ is Euler's constant i-e the limit of the series $(1+\frac12 +\frac13 +\frac14 +...+\frac1n)-\ln(n)$ where n $\rightarrow \infty$ $\gamma= 0....
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1answer
33 views

Simplify Γ(2α) in Γ(α) terms [closed]

I need to simplify Γ(2α) in Γ(α) terms where α is a real number greater than 1. (I need to cancel out Γ( . ) terms in a simplification). Shall accept a product term with Γ(α), like (2α - 1)*...αΓ(α). ...
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28 views

Closed form for an integral involving an incomplete Gamma function?

I am trying to find a closed form for this integral: $$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\...
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1answer
39 views

Definite integral inequality containing product of gamma function.

question: to prove: $I=\displaystyle\int_{0}^{1} x \Gamma\left(\dfrac{2+x}{2}\right)\Gamma\left(\dfrac{2-x}{2}\right)dx\leq\dfrac{\sqrt{6\ln2}}{3}$ my attempt: i tried using cauchy schwarz ...
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0answers
13 views

Integrate a Generalized incomplete gamma function

My question is about an integral but not an ordinary one $\int_{le^{-bt'}}^{l}s^{l-1}e^{-s}ds$ I have the idea of use the leibniz integral rule because I don't want to use the Generalized ...