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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Is it possible to calculate the area of a squircle by hand?
I got this question in an online assessment where you weren't allowed to use a programming language or online resources.
The question defines a squircle by $y^4 + x^4 = R^4$. It's area would be $4\int^...
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How to solve this integral explicitly including incomplete Gamma function?
I could not find the explicit solution for the following integral including incomplete Gamma function:$$\int_{-a}^{a} e^{At}\Gamma[0,A(t+a)]\frac{\sqrt{a^2-t^{2}}}{z-t} dt$$where $a, A>0$, $z$ is ...
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When does a line equal the Gamma function?
How would you solve for $x$ in the following equation:
$$
x = \left( x - 1 \right) ! = \int_0^\infty t^{x-1} e^{-t} dt
$$
If we are only concerned about integers, then clearly, the only solution is $1$...
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How to prove $\lim_{n\to \infty}(1+n+\frac{n^2}{2!}+\dots +\frac{n^n}{n!})e^{-n}=\frac{1}{2}$? [duplicate]
Perform a Taylor expansion of $e^x$ at $x=0$, then
$$e^n=1+n+\frac{n^2}{2!}+\dots+\frac{n^n}{n!}+\int_0^{n}\frac{(n-t)^n}{n!}e^ndt,$$
we get
$$(1+n+\frac{n^2}{2!}+\dots +\frac{n^n}{n!})e^{-n}=1-e^{-n}\...
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Proof for $\Gamma\Big(x+\frac{1}{2}\Big)^2 < x\Gamma(x)^2$
I've encountered the following inequality which I am not able to prove but pretty certain that it is true:
$$\Gamma\Big(x+\frac{1}{2}\Big)^2 < x\Gamma(x)^2$$
This should be true for $x\in\mathbb{R}^...
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Solve an equation consisting of several incomplete Gamma functions.
How should I solve this equation, please. Can the positive solution x be represented precisely by a,b, and c? If not, are there some approximations to solve the problem?
[\begin{array}{l}
\Gamma \left(...
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Can we evaluate the integral $ I(a)=\int_0^{\infty} \frac{\sin x}{x^a} e^{-x} d x, $ without Gamma functions?
Encountering the integral in the post stating that
$$
\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x}\,dx=\frac\pi4,
$$
I started to investigate the integral in a more general form as
$$
I(a)=\int_0^{\...
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The absurdity of $\Gamma(x)$'s minimum, and can it be applied to the factorial?
I know that the Gamma function can be used as a representation of the factorial, but, at the same time, it is an extrapolation of $x!$. The Gamma function is cool and all, but what are its ...
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$\infty$-multifactorial: $\displaystyle z!_{(\infty)}:=\lim_{\alpha\to\infty}z!_{(\alpha)}$
Introduction
Let $z!_{(\alpha)}$ the $\alpha$-multifactorial.
$$z!_{(\alpha)}=\alpha^{\frac{z}{\alpha}}\Gamma\left(1+\frac{z}{\alpha}\right)\prod_{j=1}^{\alpha-1}\left(\frac{\alpha^{\frac{\alpha-j}{\...
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Beta Function but with a linear term in denominator? [closed]
The integral :
$\int_0^1 \frac{t^{p-1} (1-t)^{q-1}}{(b+t)^{p+q}} dt $
I am supposed to solve this using Beta and Gamma functions, but I am not able to come to a solution, help me to find where I am ...
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Where are the zeros of $\Gamma[z,1]$?
I find a lot of literature describing $\Gamma(n,z)$ for $n$ an integer, but in this case, I care about the $n$ being complex and $z$ being fixed to $1$. According to wikipedia, it appears that $\Gamma(...
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$\displaystyle\sum_{n=1}^{\infty}{\binom{v}{n}\dfrac{\Gamma\left({\large\frac{n}{2}}\right)}{\Gamma\left({\large\frac{n+1}{2}}\right)}}$
Recently I obtained an infinite series form involving Binomial coefficient and Gamma functions when calculating the following parametric integrals:
(Tip: $\displaystyle v\ \in\ \mathbb{R}$)
$\\ \\ \...
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Prove the following Gauss formula
I was doing exercises for ahlfors's Complex Analysis book and came across the following problem
$$(2π)^{\frac{n－1}{2}}\Gamma(z)＝n^{z－\frac{1}{2}}\Gamma(\frac{z}{n})\Gamma(\frac{z＋1}{n})...\Gamma(\frac{...
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How to prove this summation equation? [duplicate]
I'm looking for some hints on proving the following (either directly or by induction):
$$
\sum_{k={0}}^{l/2} \frac{(-1)^k(2l-2k)!}{k!(l-k)!(l-2k)!} =2^l
$$
I do know it is actually true from various ...
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Asymptotic expression of the maximum of the relative error of the Fourier expansion of the multifactorial
I wanted to study the following problem:
Let
$$x!_{(a)}=a^{\frac{x}{a}}\Gamma\left(1+\frac{x}{a}\right)\prod_{j=1}^{a-1}\left(\frac{a^{\frac{a-j}{a}}}{\Gamma\left(\frac{j}{a}\right)}\right)^{C_{a}\...
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Integral of exponential function over an $n-1$ - simplex
I am trying to solve the following integral over the simplex (I'm not sure if there even is a closed form to be honest)
$$
\int_{\Delta^{n-1}}\prod\limits_{i = 1}^n x_{i}^{a_i}e^{-b_ix_i}dx_i
$$
Where ...
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How do I prove the relation: $\intop_{t=0}^{+\infty}\frac{e^{-t}}{\sqrt{t}}dt=2\sqrt{2}\intop_{x=0}^{+\infty}{\sin(x^2)}dx$
I want to prove the following relation:
$${\Gamma_{1/2}}=\intop_{t=0}^{+\infty}\frac{e^{-t}}{\sqrt{t}}dt=2\sqrt{2}\intop_{x=0}^{+\infty}{\sin(x^2)}dx$$
I noticed that: $$\frac{\intop_{x=-\infty}^{+\...
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Specific identity involving gamma function
I was trying to obtain Bailey's hypergeometric summation formula relying on this paper.
The formula is:
$$_{2}F_{1}(a,1-a;c;\frac{1}{2}) = \frac{\Gamma(\frac{c}{2})\Gamma(\frac{c+1}{2})}{\Gamma(\frac{...
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Defining $\Gamma(x)$ where $x<0$
Is it possible to define
$$
\frac{\Gamma(a+b-1)}{\Gamma(a-1)} = \frac{\Gamma(1-(a+b))}{\Gamma(1-a)}
$$
$$
s.t. a+b>0 \\ a,i\in \mathbb N
$$
I know $$\Gamma(x)$$ is defined when $$x>0,x\in \...
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Wolfram wishlist: series of $\Gamma(z)$ in $z=-m$. Cycle index of symmetric groups.
I found out that Wolfram has a wish list of formulas it is researching.
The first point is "Series for the gamma function"
We are searching for general formulas for the series expansion of ...
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Integral representation of $\psi^{(\nu)}(z)$ for $\nu>0$ but $\nu\not\in\mathbb{N}$
I need the integral representation of the polygamma function for $n>0$ but $n\not\in\mathbb{N}$.
I searched both among Wolfram functions and Digital Library of Mathematical Functions
On Wolfram I ...
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Find the exact Pochammer notation for $\binom{-2a}{n}$ and its binomial expression
Find the exact Pochammer notation for $\binom{-2a}{n}$ and its binomial expression, whch I have tried to do by the following
$$\binom{-2a}{n}=\frac{(-2a)!}{n!}=\frac{(-1)^n(2)^na(a+\frac{1}{2})(\cdots)...
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How to evaluate real and imaginary part of $\Gamma\left(\frac{2}{3},-\frac{1}{3}\right)$
I calculated the principal value of the following integral:
$$PV\int_{0}^{\infty}\frac{t^{\frac{1}{3}}e^{-t}}{1-3t}dt=\left(\frac{\Im\left[\left(-1\right)^{\frac{5}{6}}\Gamma\left(\frac{2}{3},-\frac{1}...
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Prove a certain equation [duplicate]
Can anyone prove this equation? $$\sum_{n=0}^{\infty} (-1)^n \left( \frac{(2n-1)!!}{(2n)!!} \right)^3 = \left( \frac{\Gamma\left(\frac{9}{8}\right)}{\Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{...
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clarification on gamma function to Pochhammer notation
I am going through a paper:
Nijimbere, Victor, Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals. I, Ural Math. J. 4, No. 1, 24-42 (2018). ...
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How to solve the Gamma function problem
This problem appeared in Smith's prize exam 1875.
Evaluate the modulus of
\begin{equation}
\Gamma\left(\frac{1}{2}+\sqrt{-1}a\right)
\end{equation}
If we use the corollary
\begin{equation}
\Gamma(x)\...
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Evaluate $\int_0^\infty\frac{\sin x^p}{x^p}dx$ with residue theorem
Evaluate $I=\int_0^\infty\frac{\sin x^p}{x^p}dx\,\,(p>\frac12)$.
I was able to solve it by converting the integral to a gamma function integral. We have $$I=\Im\left(\int_0^\infty\frac{e^{i x^p}}{x^...
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Show $\frac{\pi^2}{6}\sqrt{ab+\sqrt{ab+2\sqrt{ab+3\sqrt{\cdots}}}}>4$ by hand .
Problem :
Show that if :
$$a=\int_{0}^{1}x!dx,b=\int_{0}^{\infty}1/\Gamma(x)dx,x!=\Gamma(x+1)$$
Then we have :
$$S=\frac{\pi^2}{6}\sqrt{ab+\sqrt{ab+2\sqrt{ab+3\sqrt{\cdots}}}}=4.0054\cdots>4$$
...
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Integral of Binomials [closed]
Is there a way to integrate a binomial coefficient? I've tried to put the following into Wolfram Alpha, but it doesn't return a result. I have a small suspicion that I'm going a bit over my head here, ...
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About the integral $\int_0^{\pi/2} (\sin x-1)/\ln(\sin x) \mathrm{d}x$
Using the Feynman Technique,
Let $$I(a) := \int_0^{\pi/2} \frac{(\sin x)^a - 1}{\ln(\sin x)} \mathrm{d}x$$
$$I’(a)= \int_0^{\pi/2} (\sin x)^a \mathrm{d}x
= \frac{\sqrt{π}}{2} \frac{\Gamma\left(\...
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Approximation of gamma function via Riemann sums at integer points
I found something curious. We know that the gamma function is defined as
$$ \Gamma(n+1) := \int_{t=0}^\infty t^n \exp(-t) dt,$$
and it has the property that $\Gamma(n+1) = n!$ for non-negative integer ...
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How to $k$ such that $(\Gamma(n))^{-a/n} \ge k$?
Let $a\in (0, 1)$ and and $n\in\mathbb N, n\ge 2$.
I would like to find a positive constant $k$ depending only on $n$ such that
$$(\Gamma(n))^{-a/n} \ge k,$$
where $\Gamma$ denotes the Gamma function: ...
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Showing $\prod_{i=0}^{n-1} (3^2+2i)=\frac{2^{n+5}\Gamma(n+\frac{11}{2})}{105\sqrt{\pi}}$
Deduce the following equality:
$$\prod_{i=0}^{n-1} (3^2+2i) = \frac{2^{n+5} \Gamma(n+\frac{11}{2})}{105 \sqrt{\pi}}$$
So I have the product by expansion:
$$\prod_{i=0}^{n-1} (3^2+2i)=3^2(3^2+2\cdot 1)(...
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Lower bound for $\left(\frac{\Gamma(a/2)}{\Gamma(a)}\right)^{1/a}$
Let $a\in\mathbb N, a\ge 2$. During calculus class, the lecturer said that
$$\left(\frac{\Gamma(a/2)}{\Gamma(a)}\right)^{1/a}\ge \frac{1}{2a},$$
where $\Gamma$ denotes the Gamma-function: https://en....
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Express the following $\int \cdots \int \prod_i AR_i ^{\alpha_i-s_i}dR_{k-1} \cdots dR_1$ in binomial form
I have the following integral series,
$$\int \cdots \int \prod_i AR_i ^{\alpha_i-s_i}dR_{k-1} \cdots dR_1$$
Which is part of the beta theorem, and here $R_k$ stands for $1-R_1-\cdots-R_{k-1}$, so we ...
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Removing first factor of q-pochammer
What does it mean to say 'removing the first factor ' from the following function :
$$f(a)=\int_0^{\infty}t^{x-1}\frac{(-at;q)_{\infty}dt}{(-t;q)_{\infty}}$$
This interval converges when $x>0$ and $...
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Series expansion of the power Reciprocal gamma function
Based on the post Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function, is it possible to obtain some Taylor expansion series for
$$
f(z)=\frac{1}{[\Gamma(z)]...
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Upper bound for quotient of Gamma functions
Yesterday I posted a question about the Gamma function (see here: Lower bound for combined Gamma functions)
It helped me very much in solving all my remaining exercises concerning the Gamma function, ...
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1
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Lower bound for combined Gamma functions
I am asking for your help with an exercise from my general physics I class. The exercise is about the Gamma function and reads in this way.
Let $b\in\mathbb{N}, b\ge 2$ and $a\in (0, 1)$. Prove that ...
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1
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Evaluating $\sum_{k=0}^{\infty}4^{-k}\left | \Gamma(-k+\frac{i}{2}) \right |^2$
I've come across a certain hypergeometric series and have tried to express it in a different way.
So far I've got the following sum:
$$\sum_{k=0}^{\infty}4^{-k}\left | \Gamma(-k+\frac{i}{2}) \right |^...
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votes
2
answers
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Integral over a product of polynomial, exponential and Bessel function
In a physics textbook I'm working through I found an interesting integral identity which I want to prove:
\begin{equation}
\int_0^\infty t^{\nu +1} J_\nu(\beta t) e^{-\alpha t} \, dt = \frac{2\alpha (...
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0
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Series with poles of Gamma function
So I have a series:
$\sum_{n=0}^\infty \frac{x^n}{\Gamma(-n)}$.
Can I just argue that $\lim_{n\to k} \frac{x^n}{\Gamma(-n)}$, $\forall k \in N_0$ is zero and thus the whole series becomes a null ...
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Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers
As I got no answers, I reposted this question in MO.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...
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A question on Beta function
I need an asymptotic expansion/closed form for $$\sum_{k=1}^{\infty}\int_{0}^{\infty}(B(x+n+k,n+1))^2\ dx$$ where $B(m,n)$ is the Beta function and $n\in\mathbb{N}$.
Denote $$I_n=\sum_{k=1}^{\infty}\...
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What are some formulas involving Dottie number and digamma function?
Here are two formulas that involve Dottie number and Gamma function:
$\Gamma \left( \frac{1}{2} + \frac{d}{\pi} \right) \Gamma \left( \frac{1}{2} - \frac{d}{\pi} \right) = \frac{\pi}{d}
$
and
$\sqrt{\...
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2
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First order linear ODE solution
Given the following ODE
$$
2\left[1 - x - b\right]y' -2a(1-x)y + x^{a+1}=0
$$
I am trying to find a solution for $y(x)$. My approach has been to firstly rearrange the equation in the canonical form
$$
...
- 541
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0
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Interesting pattern with continued fraction and Gamma function?
As the title says I would like to know if interesting pattern occurs starting form numerical analysis with the conitnued fraction :
$$f(x)=\frac{x}{x!+\frac{2x^{2}}{x!!+\frac{3x^{3}}{x!!!+\frac{4x^{4}}...
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votes
1
answer
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Does this formula for the gamma function exist? [duplicate]
I'm a calculus student and I think I have found a new formula to express the gamma function - $\Gamma(x)$ as a limit of an infinite sum. I haven't been able to find such formulas and if somebody knows ...
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0
answers
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Derivative of the gamma function with respect to $x$ when the argument is $f(x)$?
I want to find the following derivative:
$\frac{\partial\;{\Gamma({1+\frac{\alpha}{x}})}}{\partial{x}}$, where $\Gamma(.)$ is the gamma function and $\alpha$ is a constant. If it matters, $0\leq\alpha\...
Abdahrt
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votes
1
answer
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Evaluation of a limit involving gamma function
Basically I try to answer this question. I am almost able to prove it straight on, but I hit a roadblock at the very last step. Namely, the limit
$$
\lim_{n\rightarrow \infty} \left( \sqrt[n+1]{\Gamma ...
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