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.

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Integral representation of $\frac{1}{\Gamma(z)}$

I am trying to find the integral representation of $\frac{1}{\Gamma(z)}$ in the real axis and cant seem to find it. I know that this must have a standard representation but still cant find it.
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Integral representation of modified Bessel function of second kind

I'm looking for a proof of the following integral representation for the modified Bessel function of the second kind $K_q(\rho)$ for $q \geq 0$ and $\rho>0$ $$K_q(\rho)=\frac{\Gamma(1/2)(\rho/2)^q}{...
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Is there a closed-form expression for $f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))$ assuming $x\in\mathbb{R}$?

Question: Is there a closed-form expression for $$f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))\,,\quad x\in\mathbb{R}\tag{1}$$ where $\text{Ei}(z)$ is the exponential integral ...
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Difference of modified Bessel functions in integral form

I know the following integral representation of modified Bessel functions of first kind: $$I_q(\rho) = \frac{\left(\frac{\rho}{2}\right)^q}{\Gamma(q+1/2)\Gamma(1/2)} \int_{-1}^{1}e^{-\rho t}(1-t^2)^{q-...
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Expressing $\int_0^\pi \sin^n(x)dx$ in terms of the gamma function

Let $I_n = \int_0^\pi \sin^n(x)dx$ and suppose that we have already established a recursive relation $I_n = \frac{n - 1}{n}I_{n-2}$ and we know that $\Gamma(x + 1) = x\Gamma(x), \Gamma(1/2) = \sqrt{\...
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How did Lanczos find his approximation for the Gamma function?

The Lanczos approximation gives a fixed-precision method for calculating the Gamma function. It is used in Desmos in their factorial function. According to this Wikipedia page, Lanczos derived his ...
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Asymptotic behavior of modified Bessel function help

The modified Bessel function of the first kind is defined as $$I_q(\rho)=\sum_{m=0}^{\infty} \frac{\left(\frac{\rho}{2}\right)^{2m+q}}{m!\Gamma(m+q+1)}$$ where $\rho \in \mathbb{C}\setminus \{0\}$ and ...
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Gaussian integral in 3rd dimensions

I have been wondering about computing $(1/3)!$ and using the Gamma function. After substituting for $x=t^{\frac{1}{3}}$, I got $\int_{0}^{\infty}e^{-x^{3}}dx$. May I know if there is any way to ...
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Notation: number in parentheses with subscript outside

In a reputable source I found the formula $\Gamma(z + n) = (z)_n \Gamma(z)$. What does the notation $(z)_n$ signify?
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Help in deciding if this complex function is bounded

Let $a \neq 0\in \mathbb{C}$ and consider the function $$f(z) = \frac{a^{z}}{\Gamma(z+1)}.$$ It is true that $f$ is bounded in every compact set contained in $\mathbb{C}\setminus\{-1,-2,-3,-4,\dots\}$...
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Uniform bound on product of Gamma functions in an article by Jerison and Kenig

I have been trying to read Jerison and Kenig's article Unique continuation and absence of positive eigenvalues for Schrödinger operators, and I am having difficulties understanding how they obtain the ...
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How to prove that this series of functions converges uniformly?

Let $z_0 \in \mathbb{C} \setminus \{0\}$ be a fixed complex number and let $z \in \mathbb{C}$. Let $K \subset \mathbb{C}$ be a compact set. I'm trying to use the Weiertrass $M$ - test to show that the ...
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About the inequality conjectured as $x!>\left(\arctan\left(\cosh\left(x\right)\right)\right)^{a}$ for $x>0$ and fixed $a$

I'm glad to share my last (little) discovery concerning the Gamma function or here $x!$ The problem : Let $x>0$ and $1\leq a \leq \left(\frac{\pi}{e}\right)^{e}$ then it seems we have : $$f(x)=x!&...
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Partial wave decomposition of a function with Gamma functions

I have the following function $$f(x,s) = \frac{\Gamma(1-\frac{s}{2}(x-1))}{\Gamma(1+\frac{s}{2}(x-1))}$$ where $s>0$, $x \in [-1,1] $ and $\Gamma(z)$ are Gamma functions. I would like to decompose ...
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For what reason Bessel functions of the first kind can be differenciated in relation to the variable $q$?

For that reason the modified bessel function $I_q(\rho)$ defined as $$ I_{q}(\rho)=\sum_{m=0}^{\infty}\frac{(\rho/2)^{2m+q}}{m!\Gamma(m+q+1)},$$ where in the above $\rho$ is fixed, can be ...
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"Tipping point" between asymptotic behavior of gamma function along the line $z=x+mxi$

Disclaimer: I am an undergraduate student about a semester into introductory complex analysis. I am entirely out of my depth here, just curious about something I noticed. The gamma function $\Gamma(z) ...
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6 votes
3 answers
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How to solve this limit with factorial? $\lim_{n\to \infty}\frac{n!}{n^n}(\sum_{k=0}^n\frac{n^k}{k!}-\sum_{k=n+1}^\infty \frac{n^k}{k!})$

I want to solve the limit $$\lim_{n\to \infty}\frac{n!}{n^n}\left(\sum_{k=0}^n\frac{n^k}{k!}-\sum_{k=n+1}^\infty \frac{n^k}{k!}\right)$$ This problem may be about Stirling's Approximation and Taylor ...
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Derivation of Digamma function

In the paper by Kraskov et al (2004) there is a rather large jump in calculations. I am wondering if someone could fill out the gap for the equation below (equation 17 in the paper): $$ k\binom{N-1}{k}...
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An integral equality with respect to Gamma function

How to prove that $$\int_0^{\infty}\biggl(\frac{r}{1+r^2}\biggr)^q\frac{1}{r^{1+\alpha}}dr=\frac{\Gamma(\frac{q+\alpha}{2})\Gamma(\frac{q-\alpha}{2})}{2\Gamma(q)}\quad?$$
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Help in calculation of the Wronskian of modified Bessel functions of the first kind

We define the modified Bessel function of first kind as $$I_q(z)=\sum_{m=0}^{\infty} \frac{\left(\frac{z}{2}\right)^{2m+q}}{m!\Gamma(m+q+1)}.$$ I need help in calculate the Wronskian $W(I_q(z),I_{-q}(...
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Using the Incomplete Gamma Function with real negative arguments to solve an integral

The Incomplete Gamma Function is given by $$\Gamma(a, x)=\int_x^{\infty}t^{a-1}e^tdt,$$ where $a$ and $x$ are both positive and real (for this question). My goal is to solve the following integral: $$...
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The gamma function as a limit of a lower incomplete gamma function

We define the gamma function $\Gamma : \mathbb{R}^+ \rightarrow \mathbb{R}$ as $$\Gamma(x) = \int^{\infty}_0 y^{x-1} e^{-y} dy$$ Is it possible to prove that this function can also be written as a ...
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Proof of Identity $x!=\int_{t=0}^\infty\left(-e^{-t}\sum_{n=0}^x\left(\frac{x!t^n}{n!} \right ) \right )$

Last night, I had asked a question about ${{52\choose x}\choose y}$ which led me down a rabbit hole of integration. In doing this, I discovered something for myself. $$x!=\Gamma(x+1)=\int_0^\infty\...
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Showing that $\Gamma(s+1) = s\Gamma(s)$ and the Consequences of this Identity

For $\sigma \in \mathbb{R},$ define $\Omega_{\sigma} = \{ s\in \mathbb{C}\;|\; Re(s) > \sigma\}$. On $\Omega_0$, define $\Gamma(s)$ by \begin{align*} \Gamma(s) = \int_{0}^{\infty}x^{s-1}e^{-x}dx \...
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Gamma functions approaches solely n to a negative power

I have read $[x^n] (1-x)^\alpha\sim \frac{n^{-\alpha-1}}{\Gamma(-\alpha)}$ as $n\rightarrow\infty$ where $[x^n]$ means coefficient of $x^n$ in what follows. But I have never seen a proof of this ...
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Writing factorial $x!$ in terms of the $p$-adic gamma function?

Let $x$ be a $p$adic integer. How can we write $x\cdot (x-1)\cdot (x-2)\cdots (x-n)$ in terms of the $p$-adic gamma function (Morita gamma function) ? The same question for the product $x\cdot (x-2)\...
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integral of power of $\tan(x)$ from $0$ to $\pi/2$

Prove that $$\int_{0}^{\frac{\pi}{2}} \tan^k(x)dx=\frac{\pi}{2}\sec{\frac{\pi k}{2}}$$ for $k < 1$ . I saw this problem for the case where $k=\frac{1}{2020}$ here. Then I checked a few rational ...
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Mittag-Leffler function recurrence relation

The general Mittag-Leffler function $$E_{a,b}(z)=\sum_{h=0}^{\infty}\frac{z^h}{\Gamma(ha+b)}$$ satifies the recurrence $$E_{a,b}(z)=zE_{a,b+a}(z)+\frac1{\Gamma(b)}.$$ I am having a hard time in ...
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Gamma function for real negative numbers

I'm currently working in an animation of common functions with imaginary numbers, and my last function is the gamma function. To graph the formula I need to give $2$ inputs, the real and the imaginary,...
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What is the correct representation of the generalized gamma function?

The NIST Digital Library of Mathematical Functions defines the multivariate gamma function as $$ \Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr% }\left(-\mathbf{X}\...
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Simplifying result derived from reduction formula

My question is with regards to the following integral: $$I_n =\int_0^{\pi/2} \sin^nx\,dx$$ Where $n \geq 0$. It is pretty straightforward to demonstrate using I.B.P that $$I_n = \frac{n-1}{n} I_{n-2}$$...
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Looking for upper bound for a sum over compositions

Consider the matrix \begin{bmatrix} 1 & \alpha_1 & 0 & \dots & & & 0 \\ \alpha_1 & 1 & \alpha_2 & 0 & \dots & & 0 \\ 0 & \alpha_2 & 1 & \...
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2 votes
1 answer
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Solving a Frullani integral with the incomplete gamma function

I wrote the following integral for an integration bee earlier this year, with the intended solution being to manipulate the integrand into the form of a Frullani integral (https://en.wikipedia.org/...
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Inverse Mellin transform of products of gamma functions

I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
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Factorial as a product of three consecutive natural numbers

How to find the exact number of possible ways to represent a factorial as a product of three consecutive natural numbers? $$m \cdot (m+1) \cdot (m+2) = n!$$ where $m, n \in \mathbb{N}$ I found these ...
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Solving an infinite sum of incomplete gamma with integer parameter

In trying to simplify a distribution function, I stumbled upon this infinite sum involving an (upper) Gamma function. I would believe it can be simplified further, but can't find how. $$1- \frac{(1-\...
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An integral inside an integral turns into a product of two integrals?

I am struggling to see how after substituting this: $$\frac{1}{(1+u)^{a+b}}=\frac{1}{\Gamma(a+b)}\int_{0}^{\infty}e^{-(1+u)t}t^{a+b-1}dt$$ into this: $$B(a,b)=\int_{0}^{\infty}\dfrac{u^{a-1}}{(1+u)^{a+...
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Hypergeometric Function and it's relation to Gamma function

I've been trying to solve a Quantum mechanics problem, where I have to do the integration below: $$ \int_{-\infty}^{+\infty} \frac1{(x^2+b^2)^{2n}}dx $$ with $b$ and $n$ not specified (the problem ...
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An equation involving the incomplete Gamma function

Fix $a, b>0$. For $x >0$, I need to solve the following equation in $y$: $$ \gamma(a, xy) = y^b $$ where $\gamma(s,t) = \int_0^t r^{s-1}e^{-r} dr$ is the incomplete Gamma function. How do I ...
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Series expansion in binomial coefficients

After noticing that the first three terms of OEIS A016269 coincide with those of OEIS A030053 and the fourth differs only by $1$, and working with successive approximations, I obtained the following ...
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Computing ratio $ \frac{\int_{0}^{1}\left(1-x^{k}\right)^{n} d x}{\int_{0}^{1}\left(1-x^{k}\right)^{n+1}dx} $ by other means than Beta integrals

For any positive numbers $k$ and $n$, converting the integral $$ I_{n}:=\int_{0}^{1}\left(1-x^{k}\right)^{n} d x $$ into a Beta Function by letting $y=x^{k}$, gives $d x=\frac{1}{k} y^{\frac{1}{k}-1} ...
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How to find the closed form of $\int_{0}^{\frac{\pi}{2}} \frac{d x}{\left(p \cos ^{2} x+q \sin ^{2} x+r\right)^{n}}$, where $n\in N$?

In my post, I found the integral $$\int_{0}^{\frac{\pi}{2}} \frac{d x}{\left(a^{2} \cos ^{2} x+b^{2} \sin ^{2} x\right)^{2}}=\frac{\pi\left(a^{2}+b^{2}\right)}{4 a^{3} b^{3}}$$ Then I want to find ...
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Neyman-Pearson Lemma of the Beta Distribution

$X_{1},...X_{n}$ are a random sample from the $Be(\alpha, 2)$ distribution. We wish to test $H_{0} : \alpha = \alpha_{0}$ $H_{1} : \alpha = \alpha_{1}$ Where $\alpha_{0} < \alpha_{1}$. Show that ...
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2 votes
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Desmos Factorial function

I was playing around with gamma function approximations and I was curious of which approximation Desmos uses. It extends negatives so it can’t be the Stirling formula. Does anyone know what it is?
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2 answers
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Integral representation of the power of a positive number via the gamma function.

Let $s\in(0,1)$. I managed to prove that \begin{align} \frac{1}{\lambda^s}=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}e^{-\lambda t}dt,\qquad\lambda>0. \end{align} It directly follows from the ...
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2 votes
1 answer
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Integral involving beta function (continuous version of binomial distribution)

I'm hoping someone can help me out with a problem that's consumed me for a while. I'm working with a continuous version of the binomial distribution defined for real $n > 0$ and continuous $0 < \...
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5 votes
1 answer
103 views

Integration of power law distribution and negative arguments in the lower incomplete gamma

Dear mathematical acolytes, I am working as a materials scientist and my current topic is related to some probabilistic considerations of the microstructures of metals. I have the probability of ...
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Show the limit $\lim_{\delta \to 0}\int_{\ln(1+\delta)}^{\delta} \frac{e^{-x}}{x}\,dx \to 0 $ in a proof of the Digamma function

I want to show that $$\lim_{\delta \to 0}\int_{\ln(1+\delta)}^{\delta} \frac{e^{-x}}{x}\,dx \to 0 \tag{1}$$ Intutitively I could take the limits before integration, then I would get $$\int_{0}^{0} \...
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1 vote
1 answer
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Derivation of a function missing one step in Nandi & Aich, 1994

In Nandi and Aich, (1994) Sankhya, vol. 56, p. 129-136 (accessible here), there is a density function derived. However, there is one step where I cannot figure out how it was obtained. On bottom of p. ...
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2 votes
1 answer
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Prove that $\left( \frac{x+1}{k} \right)^x>x!$ for $k<e$

I was watching a demonstration of the relation $$\left( \frac{x+1}{2} \right)^x>x!$$ for all $x > 1$ And I thought about the biggest possible denominator in the LHS such that it will be ...
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