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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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Special Functions defined by Integrals.

There was this integral which caught my attention, when I was checking out The Applications of Beta and Gamma Functions. So, how can i prove the below, using change of variable? $$\int_{0}^{1}\frac{...
2
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1answer
28 views

Convergence of Euler's definition of the Gamma function

I was reading the wikipedia article on the Gamma function and found out that the original definition of it was... quite clever actualy. Here's the article. Anyway the definition is $$ \Gamma(z) = \...
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1answer
21 views

simple question on gamma functions monotonicity

Consider $x> 1$ and $i=1,2,3...$. I do not know much about gamma functions. Is $\Gamma(x) < \Gamma(x+i), \forall i$? I know there is some property that gamma function is always increasing in $(...
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9 views

Calculate a 2-dimensional Gauss-Hermite quadrature approximation

I need to calculate an integral in which the second term is a bivariate normal density. I thought about using a 2-dimensional Gauss-Hermite quadrature. But not being familiar with the subject, I do ...
5
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1answer
61 views

Proving $\mathcal{M}\left(\sin(x)\right)(s) = \Gamma(s)\sin\left(\frac{\pi}{2}s \right)$ using Real Analysis

recently I've been investigating Mellin Transforms and this morning solved for case of $\sin(x)$ using Ramunajan's Master Theorem. I was curious if there were any Real based methods to evaluate this ...
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2answers
23 views

Integral involving Gamma distribution

I need some help with an integral. This is the solution to one of the problems I had to do. Everything is fine, but I don't understand one step: Now how is $$\int_0^\infty \frac{\beta_n^{\alpha_n+k}}...
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1answer
30 views

Analytic continuation of Gamma function and negative moments of normal distribution

I want to evaluate the divergent integral: $$\int_0^{\infty} dx\; x^{-2} e^{-x^2}$$ My plan is to calculate the following integral instead, $$ \int_0^{\infty} dx\; x^{-2g} e^{-x^2}= \Gamma\...
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37 views

What is the significance of the half derivative of $x^n$?

I have been researching about half derivatives, and the simplest version of a half derivative I found was $x^n$. However, this half derivative only works when n is a natural number. I feel like it is ...
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56 views

How to Evaluate $1-5(\frac{1}{2})^3+9(\frac{(1)(3)}{(2)(4)})^3-13(\frac{(1)(3)(5)}{(2)(4)(6)})^3+…$

I want to Evaluate $1-5(\frac{1}{2})^3+9(\frac{(1)(3)}{(2)(4)})^3-13(\frac{(1)(3)(5)}{(2)(4)(6)})^3+...$ ,I tried from arcsin(x) series and got $\frac{1-z^4}{(1+z^4)^{\frac{2}{3}}}= 1-5(\frac{1}{2})z^...
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54 views

Prove that $\frac d {dz} \ln(\Gamma(z)) ∼ \ln(z) − \frac 1{2\,z}$

What steps do I take to figure this out? I've tried using this: $z!\sim \sqrt{2\,\pi\,z} \left(\frac z e\right)^2$
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1answer
63 views

Does my definition of an exponential-like function make any sense?

The definition of the exponential function is based on an infinite series $$ e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!} $$ To make things more complicated, we could replace the factorial with the Gamma ...
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1answer
47 views

Double-precision algorithm for inverse log gamma or log factorial?

Question in a nutshell: Can anyone point me to an algorithm for computing to double-precision floating-point (roughly 16 digits) the inverse of either log gamma or log factorial? In other words, if ...
4
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1answer
80 views

Where does $\Gamma \left( \frac{3}{2} \right) = \int _0 ^{+\infty}\! \mathrm e ^{-x^2} \, \mathrm d x$ come from?

My teacher solved this problem in class but I don't get how one step is justified. Prove that $$\int_0 ^{+\infty} \! \mathrm e ^{- x^2 } \, \mathrm d x = \dfrac{\sqrt{\pi}}{2}$$ using this ...
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1answer
36 views

Looking for an explanation or insight on a form of the inverse of a restricted gamma function

I posted this yesterday asking about how to find an inverse of a restricted gamma function. To put it concisely, I was looking at the gamma function in the positive reals after restricting it to be ...
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3answers
78 views

General formula for $\int_0^{\pi/2} \tan^{\alpha}(x) dx$?

There are already questions about how to find $\int \tan^{1/2}(x) dx$. But how to derive a general formula for $\int_0^{\pi/2} \tan^{\alpha}(x) dx$ (which converges if $|\alpha|<1$) ? More details:...
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2answers
36 views

Finding an Inverse of Restricted Gamma Function

I don't know/haven't used LaTeX yet but I'll do my best to keep it simple, I'm working on my undergrad senior project and I'm trying to find an inverse function for f(x)=(x-1)! just in the positive ...
4
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1answer
81 views

Lower bound for $\Gamma(x+i y)$ where $x>0$ involving only the real part

I am trying to have some inequalities involving the special Gamma function. I am able to get an upper bound for $\Gamma(x+i y)$, for $x>0$, $$ \begin{align} |\Gamma(x+iy)| &=\left|\int_0^\...
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0answers
21 views

zeros of imaginary part of $\log(\Gamma(z))$

I want to discover if there exist a solution of the following equation: $$\Im(\log(\Gamma(iy)) - y\beta = 0 $$ for $y\in\mathbb{R}\setminus \{0\}$ and $\beta >0$ I already proved that if there is ...
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1answer
60 views

Integral involving reciprocal gamma function

I'm interested to find the value of the following integral involving the reprocal gamma function $$\int_0^{\infty}\frac{(u+\beta)^n}{\Gamma(u+\alpha)}du$$ where $\alpha, \beta>0$ (can be the same)...
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Evaluate $\sum\limits_{k=1}^{\infty}\frac{B(k,k)}{k}$ where $B$ denotes the beta function

How can I evaluate $\displaystyle\sum_{k=1}^{\infty}\frac{B(k,k)}{k}$ Here B is the beta function
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1answer
60 views

Can someone clearly explain the algebra involved in this sum and integral?

We have : $$\int \frac{1}{x} \sum_{m=1}^\infty \frac{(x y)^m}{1-y^m} dx = \sum_{m=1}^\infty \frac{y^m}{1-y^m} \int x^{m-1}dx= \sum_{m=1}^\infty \frac{y^m}{1-y^m} \frac{x^m}{m} = \sum_{m=1}^\infty \...
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1answer
15 views

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family.

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family. I know that the beta distribution is $f(x; \alpha, \beta)={1\over B(\alpha, \beta)}x^{\...
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0answers
20 views

Calculating double integral with incomplete gamma function (numerically)

I need to calculate a double integral on two variables ($b_0$ and $b_1$) in a function that includes a gamma incomplete function such as : $\int_{\Bbb R} \frac{1}{\eta} (\lambda e^{b_{0} + b_{1}z_{1}}...
3
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2answers
63 views

how to find $\Gamma(n+3/2)$

I'm newly introduced to the gamma function. I was wondering how can I calculate: $$\left(n + \frac 12\right)!$$ When I entered the above in wolfram alpha the result was: $$\Gamma\left(n + \frac 32\...
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0answers
41 views

Where does the gamma function come from [duplicate]

It has long been known that the gamma function is an extension (shift) of the factorial function defined on integers. There were an infinite numbers of ways to continue the factorial, so what ...
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1answer
18 views

How to find the new parameters of a gamma distribution after scaling?

I understand this question has been asked a couple of times, but from what I am seeing, the parameters shown are different so I am not sure if I am on the right track. My question goes like this: ...
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0answers
30 views

Incomplete regularized Gamma function recursive relation

I'm trying to found the detailed derivation of: $Q(\alpha + n, t) = Q(\alpha, t) + t^\alpha e^{-t} \displaystyle\sum_{k=0}^{n-1} \dfrac{t^k}{\Gamma(\alpha+k+1)}$ This is a relation necessary for the ...
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3answers
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$\Gamma(s)= \lim_{n \to \infty} \frac{n^s n!}{s(s+1)…(s+n)}$ , the product formula of Gamma function .

Prove that $$\Gamma(s)= \lim_{n \to \infty} \frac{n^s n!}{s(s+1)...(s+n)}$$ Whenever $s \neq 0,-1,-2,...$ My attempt : By applying product formula for $\frac{1}{\Gamma}$ , $$\Gamma(s)=\lim_{n\...
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44 views

What is the Laplace transform of $\frac{\Gamma(t,z)}{\Gamma(t)}$

What is the Laplace transform of : $$\frac{\Gamma(t,z)}{\Gamma(t)}$$ w.r.t. the real parameter $t$, Where $\Gamma(\cdot,\cdot)$ is the incomplete gamma function, and $z\in \mathbb{C}$ EDIT From ...
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3answers
26 views

On an identity regarding the gamma function

On page 9 of H. Edwards' Riemann's Zeta Function, it says: Substitution of $nx$ for $x$ in Euler's integral for $\Pi(s-1)$ gives $$\int_0^\infty e^{-nx}x^{s-1}dx={\Pi(s-1)\over n^s}$$ ($s>0$, $n=...
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4answers
93 views

Asymptotic approximation regarding the Gamma function $\Gamma$.

On the wikipedia page for Gamma function I saw an interesting formula $$ \lim_{n\to \infty} \frac{\Gamma(n+\alpha)}{\Gamma(n)n^\alpha} = 1 $$ for all $\alpha\in\Bbb C$. I couldn't find the source of ...
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1answer
61 views

Representation of Gamma function on $0 < \operatorname{Re } z < 1$

I've read that for $0 < \operatorname{Re }z < 1$ the Gamma function has the following representation: $$\Gamma(z) = e^{i\pi \frac{z}{2}} \int_0^\infty t^{z-1} e^{-it} \, dt.$$ I couldn't find ...
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44 views

How to show: $\psi^{(0)}\left(\frac{1}{n}\right) - \psi^{(0)}\left(1 - \frac{1}{n}\right) = -\pi\cot\left(\frac{\pi}{n}\right)$ [duplicate]

Based on a result I found recently and in conjunction with methods I've observed on MSE I was able to show that: \begin{equation} \int_0^\infty \frac{ \ln(t)}{t^n + 1}\:dt = -\frac{\pi^2}{n^2} \...
1
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2answers
48 views

Prove that $\sqrt{\frac{n-1}2} \frac{\Gamma\left[\frac{n-1}2\right]}{\Gamma\left[\frac{n}2\right]}\gt1\quad\forall n \ge 2,n\in\mathbb N$

How to prove $$\sqrt{\frac{n-1}{2}} \frac{\Gamma\left[\frac{n-1}{2}\right]}{\Gamma\left[\frac{n}{2}\right]} \gt 1 \quad \forall n \ge 2,n\in \mathbb{N}$$ I plot this function in Mathematica and ...
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3answers
75 views

A finite summation of double binomial coefficients

I find the following identity and have checked on Mathematica, while I have no idea how to prove it: $$\sum_{j=0}^n(-1)^{n-j}\binom{p+j}{j}\binom{n+\beta}{n-j}=\binom{p-\beta}{n}, \quad \beta>-1, \...
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0answers
17 views

Can the series expansion of a convergent integral be divergent?

This is the screenshot from Arfken and Weber. The series expansion for the given integral diverges for all x. If I check on a calculator, it gives $I(5,1)=0.001148$. It means the integral converges ...
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1answer
23 views

Find the PDF of gamma distributed random variable using derivation

Let $X$ be a random variable with CDF $F_X(x)$ given by $$ F_X(x)=1-\frac{\Gamma(m,(m/y)x)}{\Gamma(m)}, $$ where $m$ and $y$ are positive integers $(m>0, y>0)$ and $\Gamma(a,z)$ is the ...
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4answers
193 views

Methods to attack integrals that include $(1+x)^{a}\ln^{b}(1+x)$ in the integrand

I am looking for systematic methods to attack the following class of integrals involving logarithmic functions $$\begin{aligned} I_{0} &= \int_{0}^{1}(1+x)^{a}\ln^{m}(1+x)\,\mathrm{d}x \\ I_{1} ...
11
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1answer
238 views

Solving $\int_0^{\frac{\pi}{2}}\frac{1}{\sin^{2n}(x) + \cos^{2n}(x)}\:dx$

Spurred on this question I decided to investigate the following integral: \begin{equation} I_n = \int_0^{\frac{\pi}{2}}\frac{1}{\sin^{2n}(x) + \cos^{2n}(x)}\:dx \end{equation} Where $n \in \mathbb{...
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1answer
41 views

Integrating $x^c (1-x)^d$ from 0 to 1 using the gamma function

I am trying to solve the following integral: $$ \int_0^1 x^c (1-x)^d dx $$ for some $c, d \in \mathbb{R}$ I know I have to use the gamma function, I have tried using the substitutions $u = ln\frac{1}{...
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0answers
39 views

Finding conditions to an equality in an integral

First of all, a happy new year to everybody. Well, I'm working on a problem. That problem stated that I've to find the conditions for which this equality holds: $$\mathcal{I}_\text{n}\left(\epsilon\...
3
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1answer
64 views

Compute $\int_{0}^{+\infty} \frac{\sin x}{\sqrt{x}}\ dx$ using Gamma function

I want to compute $$\int_{0}^{+\infty} \frac{\sin x}{\sqrt{x}}\ dx$$ using Gamma function. I know that by change of variable, $y=\sqrt{x}$, one gets $$\int_{0}^{+\infty} \frac{\sin x}{\sqrt{x}}\ ...
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2answers
97 views

Evaluate integral of $\ln(u)\exp\{-bu\}/u du$

While doing my research, I came across this integral and don't know how to solve for this: $$\int_{0}^{\infty}x^2\exp\{ax-be^{ax}\}dx,\text{where $a,b>0$}.$$ My attempt: \begin{align} \int_{0}^{\...
2
votes
1answer
73 views

Gamma function related?

I am just wondering about an integral, which has the form $$\int_0^\infty\frac{dx}{\sqrt{x(x^2+ax+1)}}$$ which also equivalent to $$2\int_0^\infty\frac{dx}{\sqrt{x^4+ax^2+1}}$$ do these Integrals have ...
5
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1answer
137 views

Solving an integral equation with inverse Laplace transform

Let $\alpha,\beta,\mu>0$. I am looking for a solution, i.e. a function $g(x)$, that satisfies $$ \frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_0^\infty g(x)x^{\alpha-1}e^{-\beta x}\,\mathrm dx=\left(\...
3
votes
2answers
133 views

How to show that $ \prod\limits_{r=1}^{n}\Gamma{ \left({\frac {r}{n+1}}\right)}={\sqrt {\frac {(2\pi )^{n}}{n+1}}}$?

I saw the following equation on Wikipedia, but I am not sure how to approach it. $$ \prod_{r=1}^{n}\Gamma{ \left({\frac {r}{n+1}}\right)}={\sqrt {\frac {(2\pi )^{n}}{n+1}}}$$ Here are some other ...
0
votes
1answer
35 views

Seeking proofs of a limit expression of the gamma function

I just started reading H. M. Edwards' Riemann's Zeta Function and I encountered an equation that I don't know how to prove. It's Equation (3) on page 8: $$\Pi(s)=\lim_{N\to\infty}\frac{1·2\cdots N}{(s+...
1
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3answers
50 views

Problem with evaluating $\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$ using Beta Function

Recently I've been trying to tackle the integral $\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$ using the Beta function $$\frac{B(\frac{x}{2},\frac{1}{2})}{2}=\int_0^{\frac{\pi}{2}} \sin^{x-1}(\...
1
vote
0answers
59 views

Solving $\Re\left[\Gamma(n, bi)\right]$

I recently was able to show: \begin{equation} \int_{0}^{\infty} \frac{e^{-kx^n}}{x^n + a}\:dx = e^{ak}a^{\frac{1}{n} - 1}\frac{\Gamma\left(\frac{1}{n} \right)\Gamma\left(1 - \frac{1}{n}, ak \right)}{...
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0answers
81 views

Is $\Gamma{(\frac{1}{5})}$ transcendental?

Wolfram Mathworld lists several transcendental numbers such as $$\Gamma{\left(\frac{1}{3}\right)},\Gamma{\left(\frac{1}{4}\right)},\Gamma{\left(\frac{1}{6}\right)}$$ I don't see the reason why Wolfram ...