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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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Beta distribution: find the parameter $\alpha$ of $\mathcal{B}e(\alpha,\frac{1}{3})$

I have this variable with beta distribution : $Y \sim \mathcal{B}e(\alpha,\frac{1}{3})$. I have to find the value of $\alpha$ such as : $P(Y \leq 0.416) =0.2 $ Formally for $\alpha \geq 0$ , $\beta \...
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Calculate $E(Y)$ where $Y=X^{1.5}$

Let $X$ be a rv which is $Exp(\lambda=2)$. The pdf of $X$ is given by $f_X(x)=2e^{-2x}, x\geq 0$ (and $0$ otherwise). We define $Y=X^{1.5}$ and ask $E(Y)$. It does not look like the moment ...
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definite integration - solution breakdown

$$ \int_0^{\infty} xe^{-x(y+1)}dx$$ $$=-\frac x {y+1}e^{-x(y+1)}|_0^{\infty} +\frac 1 {y+1} \int_0^{\infty}e^{-x(y+1)}dx$$ $$=\frac 1{(y+1)^{2}}.$$ Unfortunately i cannot follow the steps. I guess ...
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42 views

Prove $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\ln{(2n+1)}}{2n+1}=\pi/4(\gamma-\ln{\pi})+\pi\ln{(\Gamma(3/4))}$

In the title, $\gamma$ is the Euler-Mascheroni constant and $\Gamma(3/4)$ represents the extension of the factorial function. This isn't a homework question or something, someone left it on a board ...
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1answer
67 views

Prove that $\prod_{r=1}^m \sin \left( \frac {r\pi}{2m+1}\right) =\frac {\sqrt {2m+1}}{2^m}$

Prove that $$\prod_{r=1}^m \sin \left( \frac {r\pi}{2m+1}\right) =\frac {\sqrt {2m+1}}{2^m}$$ My try: $$\prod_{r=1}^m \sin \left( \frac {r\pi}{2m+1}\right) =\prod_{r=1}^m \left(\frac {e^{\frac {ir\...
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1answer
30 views

Area under the graph of $r\mapsto\binom nr$

The question: Given $n$ is a natural number and $r$ is varying from $0$ to $n$, find the area under the graph of $r\mapsto\binom nr$, taking the $\Gamma$-function definition of factorial. ...
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Irrationality of :$ \zeta \left(\frac{1}{\phi}\right)\Gamma{\left(\frac{1}{\phi}\right)}$

This number : $$ \zeta \left(\frac{1}{\phi}\right)\Gamma{\left(\frac{1}{\phi}\right)}$$ almost integer and it's close to $-3$ with $\phi$ is the Golden ratio and Zeta is the Riemann zeta function , $\...
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On Ramanujan's proof of Bertrand's postulate (using stirling formula)

I'm trying to understand Ramanujan's proof of Bertrand's postulate, but I don't get the step in which it says But is easy to see that $\log\Gamma(x) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2}) \...
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How to show that $\left| \Gamma \left(x + iy \right) \right|^{2} \approx (\pi y^{(2x - 1)}) /(\cosh(\pi y))$?

I found the approximation: $$\left| \Gamma \left(x + iy \right) \right|^{2} \approx \frac{\pi y^{(2x - 1)}}{\cosh(\pi y)} $$ for $y \gt 2$, within an answer for another question, but I could not ...
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A More Direct Proof Please of a Theorem of the Incomplete Gamma Function

I could not find a tag for incomplete gamma function, so I've just used the gamma function one. A rather curious fact about the normalised incomplete gamma function$${1\over n!}\operatorname{\Gamma}(...
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56 views

On an Inequality for the Riemann Zeta Function

Okay, firstly a bit of background to set the scene. My question comes from the approaches made by R. Spira in his paper, "An inequality for the riemann zeta function," regarding the initial steps he ...
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Does the integral of the gamma function ever occur atall?

Does the integral of the gamma function ever occur in any department of mathematics? And I am not talking about integrals inwhich you have the product of two gamma functions facing in opposite ...
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How do I prove $z\Delta(z+1) = \Delta(z)$, where $\Gamma(z)=1/\Delta(z)$?

EDIT notes: I found a mistake in my calculations. * something * means that this term should be omitted (I'm not going to delete those terms, since my question would become meaningless) I'm asked to ...
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45 views

Can $\int_{-\pi}^\pi e^{\cos(\theta)}\sin(\sin(\theta))\sin(nx)dx=\frac{\pi}{n!}$ be extened to non-integers?

It is well known that the fourier series for $e^{\cos(\theta)}\sin(\sin(\theta))$ is $\sum_{n=1}^\infty \frac{\sin(n\theta)}{n!}$ which implies that $$\int_{-\pi}^\pi e^{\cos(\theta)}\sin(\sin(\theta)...
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135 views

Calculating $S=\sum\limits_{n=1}^\infty\left(\frac{1}{\Gamma^2(n+1)}\right)^{{1}/{n}}$

I tried to find the answer for the question: Numerical evaluation of $\sum_{N=1}^\infty\left(\frac{1}{\Gamma(N+1)^2}\right)^{\frac{1}{N}}$. I think my result is $4$ times than the expected value. Is ...
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Evaluation of :$\sum_{n\geq 2}\frac1n\Gamma(\frac1n)^{\zeta{(\frac1n)}}$

I want to know more about behavior of both Gamma function and zeta function writing them as a power in the form of harmonic series which i got the below form $$\sum_{n\geq 2}\dfrac1n\left(\Gamma\left(\...
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Numerical evaluation of $\sum_{N=1}^\infty\left(\frac{1}{\Gamma(N+1)^2}\right)^{\frac{1}{N}}$

Given $$S=\sum_{N=1}^\infty\left(\dfrac{1}{\Gamma(N+1)^2}\right)^{\dfrac{1}{N}}$$ Using the Carleman inequality, I got for S: $$S\le\dfrac{1}{6}e\pi^2$$ Using numerical calculation I suppose that the ...
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3answers
55 views

Calculate limit of \Gamma function

Show that $$\lim _{x \to \infty} \log \left( \frac{ \sqrt{x} \Gamma\left(\frac{x}{2}\right) } {\Gamma \left( \frac{x+1}{2}\right)} \right) = \frac{1}{2} \log(2),$$ where $\Gamma$ is the Gamma ...
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1answer
40 views

Integral involving the log gamma function

I have used the Kummer representation series of loggamma function but does not look promissing to tackle this integral. Any idea to calculate this integral in closed-form ? $$\int_{0}^{1}\ln(x)\ln\...
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685 views

How to evaluate this nonelementary integral?

Let $x>0$. I have to prove that $$ \int_{0}^{\infty}\frac{\cos x}{x^p}dx=\frac{\pi}{2\Gamma(p)\cos(p\frac{\pi}{2})}\tag{1} $$ by converting the integral on the left side to a double integral ...
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1answer
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how to choose the sabstitution for Euler's integrals?

I currently have, and have to calculate the Gamma function: $$\int_2^4 \sqrt[4]{(x-2)(4-x)^3}\,\mathbb{d}x$$ As per definition gamma function is: $$\int_0^1t^{z-1}e^{-t}\,\mathbb{d}t$$ Do I ...
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2answers
37 views

integrating gamma pdf over fixed limits

I am trying to solve $\int \limits _u^v x^{m-1}e^{-x} dx$. I checked table of integrals too but there is no direct solution for this, any help?
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1answer
28 views

Polygamma expression for $\frac{\Gamma^{(k)}(z)}{\Gamma(z)}$?

I'm trying to simplify $$\frac{\Gamma^{(k)}(z)}{\Gamma(z)}$$ for $k=1,2,\cdots$, using polygamma notation Try I've calculated a few, using $$\Gamma^{(k)}(z) = \int_0^\infty (\log x)^k x^{z-1} ...
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0answers
34 views

Does convergence of $\Gamma(x)$ imply convergence of $\Gamma(z)$? Is it generalisable?

If we have the gamma function in integral form $\Gamma(x)=\int_\limits0^\infty e^{-t}t^{x-1}dt$ and have proven that it converges for real $x>0$ (to a point), can we then immediately conclude that $...
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41 views

How to show $\frac{\Gamma((n-1)/2)}{\Gamma(n/2)} \approx \frac{\sqrt{2}}{\sqrt{n-2}}$

Show $\frac{\Gamma((n-1)/2)}{\Gamma(n/2)} \approx \frac{\sqrt{2}}{\sqrt{n-2}}$ Try Using the facts: $(1 + \alpha/m)^m = e^\alpha ( 1+ r_m)$, where $\lim_{m \to \infty} \sqrt{m}r_m = 0$ $\Gamma(n+1) ...
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Gamma function limt to integral question

I am reading up on the gamma function and have seen a formula that I can't connect to the usual integral definition. Namely, $$ \Gamma(x) = \lim_{n\rightarrow \infty}\frac{n!n^{x-1}}{x(x+1)\cdots(x+...
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1answer
42 views

Evaluating product of Upper Incomplete Gamma functions

I have checked several posts but couldn't find the equivalent of $\Gamma(m,a) \cdot \Gamma(m,b)$, where '$\cdot$' means multiplication. I suspect that it can be solved by applying the equivalent of ...
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On the regularized gamma function (analysis problem)

I have a quick question on the regularized gamma function, defined as $Q (a,z)=\frac{\Gamma(a,z)}{\Gamma(a)} \,. [1]$ What is the value of $Q (a,z)$ in the asymptotic limits $a \rightarrow \infty$ ...
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67 views

large x and small x expansion for gamma-like function

Find two approximations for the integral ($x>0$) \begin{equation} I(x) = \frac{1}{2\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{x \cos^2(\theta)}d\theta \end{equation} one for small ...
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Formula for the ratio $\frac{\Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n + 1)}$ of two values of the Gamma function

Show that $$\frac{\Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n + 1)} = \frac{1 \cdot 3 \cdot \cdots (2 n - 3) (2 n - 1)}{2 \cdot 4 \cdot \cdots (2 n - 2) \cdot 2n} .$$ I have proved that $$\Gamma\...
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Integration of $g(x) = \begin{cases} x^{1/c-1}e^{-x^{-c}} &x>0\ \ \\ 0 &\text{else} \end{cases} \text{and} \ c\neq 0$

I need help to integrate this function: $$\begin{align} &g(x) = \begin{cases} x^{1/c-1}e^{-x^{-c}} \ &x>0\ \ \\ 0 &\text{else} \end{cases}\\[5pt] &\text{and} \ c\neq 0 \end{align}...
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Methods to solve $\int_{0}^{\infty} x^{n}\cos(x)\:dx$

I've been playing around with the following integral and was wondering if it can be generalised to any Real $n$. Does anyone know of any methods to approach this one? $$ I = \int_{0}^{\infty} x^n \...
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1answer
52 views

What is equivalent of $\Gamma(n/2)$?

I should solve $\pi^{n/2} / \Gamma(n/2 + 1) = 1$. Therefore, I need to know other forms of $\Gamma(n/2)$ or $(n/2)!$. I have already checked the Mathematica and MathWorld, very well. But unfortunately,...
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1answer
20 views

Can this expression arising from the Weibull distribution be further simplified?

An estimator for the shape parameter for the Weibull distribution is derived from the relation: $\displaystyle{\frac{\sigma^2}{\mu^2}} = \displaystyle{\frac{\Gamma\left(1+\frac{2}{k}\right)}{\Gamma\...
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2answers
140 views

Question about a function that is a ratio of gamma functions and appears to be strictly increasing for $x\ge 2$

I was surprised to discover that the following function appears to be strictly increasing for $x \ge 2$: $$f(x) = \frac{\Gamma(x+1)}{\Gamma(\frac{x}{2}+1)\Gamma(\frac{x}{3}+1)\Gamma(\frac{x}{5}+1)}$$ ...
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154 views

Growth of Digamma function

For $1\le \sigma \le 2$ and $t\ge 2$, $s=\sigma+it$ prove that $\displaystyle \frac{\Gamma'(s)}{\Gamma(s)}=O(\log t)$. From Stirling's formula we have, $\displaystyle \Gamma(s)\approx \sqrt{2\pi}\exp\...
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Closed form for product over Gamma function

Is there a "closed form" (with which I mean an expression not involving an indexed sum or product) for any of these four products? $$\prod_{k=1}^{n} \Gamma(\frac{x}{k*2+1})$$ $$\prod_{k=1}^{n} \Gamma(...
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inverse Laplace transform of exponential function and gamma function

I came to this final problem to be solved. I would like to understand a way to tackle this problem: Inverse Laplace transform of $$\frac{e^{-as}}{Γ(bs+c)}$$ I cannot find an easy way of finding an ...
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323 views

How to calculate the Kampé de Fériet function?

This is a continuation of this post. The following is my original question in that post. Question: Is it possible to express $$\sum_{l=0}^\infty \sum_{r=0}^\infty\frac{\Gamma(L+r-2q)}{\Gamma(L+...
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45 views

Summation of fractions of Gamma functions

Recently I gave Mathematica the following input on the left hand side $$ \sum_{n=0}^{\infty}\frac{\Gamma(a+n)}{\Gamma(b+n)}=\frac{\Gamma(a)\Gamma(b-a-1)}{\Gamma(b-1)\Gamma(b-a)}. $$ Can anyone ...
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The derivative of the Gamma function, once more

(In connection with this. My previous question was answered, and here is a modified version of my question.) The Gamma function satisfies the relation $z\Gamma(z)=\Gamma(z+1)$, whence $|\Gamma(z+1)|&...
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The derivative of the Gamma function

The Gamma function satisfies the relation $z\Gamma(z)=\Gamma(z+1)$, whence $|\Gamma(z+1)|>|\Gamma(z)|$ whenever $|z|>1$ (and $z$ is not a non-positive integer). This naturally leads us to the ...
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2answers
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If $a,b,c$ are positive integers with $c\leq a+b,$ can I conclude that $_2F_1(a,b;c;1)$ diverges?

Recall that the hypegeometric series is defined by $$_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$$ where $z\in \mathbb{C}$ with $|z|<1$ and $(a)_n = a(a+1)...(a+n-1)...
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Order of $\Gamma(n/2)$

I want to estimate the order of $\Gamma(n/2)$. We have from Stirling interpolation , for sufficiently large value of $n$, \begin{align} \Gamma(n/2)&\approx \sqrt{\frac{4\pi}{n}}\left(\frac{n}{2e}...
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1answer
32 views

Gamma Function Holomorphic

Gamma Function Holomorphic We consider a proof that shows that the Gamma function $\Gamma(z):= \int_0 ^{\infty}t^{z-1}e^{-t}dt$ is holomorphic on the complex halfplane $\{z \in \mathbb{C} \vert Re(z) ...
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1answer
45 views

Asymptotic behavior of a certain function

Let $a$ and $b$ be two positive real numbers and define $$ f(n)=\binom{2n-2}{n-1}\frac{b^{2n-2}}{(2a)^{2n+1}}\left(1-e^{-2a}\sum_{k=0}^{2n-2}\frac{(2a)^k}{k!}\right), \ \ \ n\in\mathbb{N}_{\ge 1}. $$ ...
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0answers
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Residues of the Gamma function

I am trying to make sense of a proof that the poles of $\Gamma(z)$ are at $z=-n$ and have residue $\frac{(-1)^n}{n}$. The proof reduces $\Gamma(z)$ to the sum of an (entire) incomplete gamma function ...
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1answer
75 views

How do we prove that $\int_{0}^{\pi/2}\sin(t)^{2n+3}dt=\frac{4^n(2n+2)}{(2n+3)(2n+1){2n\choose n}}$?

I saw this integral in a paper on hypergeometric functions: $$S(n)=\int_{0}^{\pi/2}\sin(t)^{2n+3}dt=\frac{4^n(2n+2)}{(2n+3)(2n+1){2n\choose n}}\;\;\;\;\;\;\;\;\;\;\;(1)$$ I tried to prove it and got ...
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1answer
31 views

How Fast Does Incomplete Gamma Converging to Gamma?

In interested in how fast $\Gamma(\alpha,x) \to \Gamma(\alpha)$ as $x \to 0$, for some fixed $\alpha>1$. For example, set $\beta>0$ and consider the limit $$\lim_{x\to0}x^{-\beta}(\Gamma(\alpha,...
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1answer
69 views

Integration problem related to Gamma function: $ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du $

During my work on some statistics problem, I stumbled across the following integral: $$ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du,\qquad \alpha, b, c>0 $$ I tried ...