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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. The Gamma function is a specific way to extend the factorial function to other values using integrals.

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On moment generating function of generalized gamma distribution

I'm reading An Intermediate Course in Probability by Gut. I am confused about a statement made concerning the generalized gamma distribution and its existence of a moment generating function. I quote: ...
psie's user avatar
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2 votes
1 answer
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Estimation of a gamma function-like integral

A random variable $X$ has a pdf: $$f(x) = \frac{1}{k!} \cdot x^k \cdot e^{-x}$$ Prove that $$P(0<X<2\cdot(k+1)) > \frac{k}{k+1}$$ There are no conditions about $k$, so it can be non-integer. ...
Disciple's user avatar
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1 answer
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Can the gamma function be generalized to quaternions and how? [duplicate]

The gamma function is a generalization of the operator !n. The question is: Can the concept of the gamma function be generalized to quaternion analysis and the use of quaternions, and how?
Anas Khallouf's user avatar
-2 votes
1 answer
106 views

Is there an analog for factorials in division, and if so, what are its applications and properties? [closed]

If we consider a factorial to be an operation/function of iterative multiplication, would it be reasonable to think that something similar for division also exists? If we take this function to be f, ...
Pratixit Tripathy's user avatar
3 votes
1 answer
99 views

Contour integration of an integral

I am trying to determine the following integral by using the residue theorem $$\int_{ - \infty }^\infty {\frac{{\gamma \left( {1/2,{\rm{i}}z} \right)\gamma \left( {1/2, - {\rm{i}}z} \right)}}{{{z^2} +...
Eric's user avatar
  • 31
2 votes
3 answers
88 views

How can I simplify $\prod_\limits{n=1\atop n\ne m}^{a}\frac{nx - 1}{n - m}:\ ? $

I ran into the following product while doing a Lagrange Polynomial Interpolation for a math puzzle I created for myself and am struggling to simplify it further: $$ \prod_{\Large n = 1\atop \Large n \...
Dylan Levine's user avatar
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2 votes
0 answers
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How to define double factorial for non positive integers?

I studied double factorial which known for natural number $$ n!!=n(n-2)!! , 1!!=0!!=1$$ So we have for $n\in N$ $$ (2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$ but I found on Math-World formula ...
Faoler's user avatar
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3 votes
0 answers
114 views

How to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$

In the calculation of a problem, I need to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$ holds for any positive integer $n$. I got the expansion ...
Jun Wang's user avatar
4 votes
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82 views

Is the $d$-dimensional harmonic series is proportional to the surface-area of a $d$-sphere?

TLDR: How to Prove: $$ \sum_{\sqrt{a_0^2 + ...+ a_k^2} \le n, (a_0 ,...,a_k) \ne (0)^{k+1} } \frac{1}{(a_0^2 + ...+ a_k^2)^{\frac{k}{2}}} = \frac{k\pi^{\frac{k}{2}}}{\Gamma(\frac{k}{2} + 1)} \ln(n) + ...
Sidharth Ghoshal's user avatar
0 votes
0 answers
116 views

Prove that this limit is equal to $\sqrt{2}$ for the function $f(x)=x^2-2$ for an arbitrary seed point $s$.

Mathematica knows that: $$ s + \frac{1}{1-\lim_\limits{n\ \to\ \infty}\left[\frac{\displaystyle\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{f\left(k/n + s -1/n\right)}}{\displaystyle\sum _{k=1}^...
Mats Granvik's user avatar
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0 answers
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how to use Gauss Multiplication Formula for Gamma function?

I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$ $$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)$$ but I didn't ...
Faoler's user avatar
  • 1,647
2 votes
1 answer
56 views

How can we calculate this integral if this diverges?

I know that: $$\int_{0}^{\infty} \frac{x^{s-1}}{1+x} \, dx=\Gamma(s)\Gamma(1-s) $$ where $s \notin \mathbb{Z} $ Which we know is the Euler reflection formula for the Gamma function and evaluates to :$$...
Prince Yadav's user avatar
0 votes
0 answers
42 views

Symbolic differentiation Gamma function (rings - algebra)

I was after an expression for the the nth derivative of the Gamma function and I managed to find this Symbolic differentiation Gamma which reads as \begin{equation} \Gamma^{(n)}(z) = \Gamma(z) R(n,z) \...
FM89's user avatar
  • 13
10 votes
2 answers
405 views

How to prove the following integral inequality problem

I have a question about an inequality I am trying to integrate. $$\int_0^1 (1 - x^a)^b ~dx \ge \left(1-\frac{1}{a+1}\right)\frac{1}{b^{\frac{1}{a}}},$$ where $a, b \ge 1$ are integers. Any assistance ...
Lanchao Wang's user avatar
2 votes
2 answers
102 views

I need help to prove that :$\sum_{n=0}^{\infty}\frac{\Gamma^{2}(n−\frac{1}{2})}{\Gamma^{2}(n+1)}=16$

$$ \mbox{I need help to prove that:}\quad \sum_{n = 0}^{\infty}\frac{\Gamma^{2}\left(n − 1/2\right)}{\Gamma^{2}\left(n + 1\right)}=16 $$ The gamma function is a special function that I don't know a ...
Mostafa's user avatar
  • 2,348
4 votes
2 answers
97 views

Estimates for incomplete Gamma functions

I want to show that $$\int_{0}^{\sqrt{n}} \exp(-s^4)s^{n+1} ds - \int_{\sqrt{n}}^{\infty} \exp(-s^4)s^{n+1} ds \geq 0\, \quad \text{for all }n\in\mathbb{N}.$$ These integrals can in fact be written ...
Nils's user avatar
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8 votes
3 answers
235 views

How to calculate $\int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x$

As the title mentioned, I want to calculate \begin{equation} \int_{0}^{\infty} x^n(1+x)^n e^{-ncx^2}\text{d}x, \end{equation} where $n$ is a positive integer, $c$ is a positive real number in the ...
Jobs Adam's user avatar
  • 243
0 votes
0 answers
26 views

Question on convergence of Fourier transform of $f_a(t)=\Gamma(a+i t)$ where $a\in\mathbb{R}$

Mathematica indicates the inverse Fourier transform of $F(\omega)=e^{-\omega-e^{-\omega}}$ is $$f(t)=\mathcal{F}_{\omega}^{-1}\left[e^{-\omega-e^{-\omega}}\right](t)=\int\limits_{-\infty}^{\infty} e^{-...
Steven Clark's user avatar
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7 votes
2 answers
213 views

The sum $s(k)=\sum_{n=1}^\infty\frac{\Gamma(\frac{1}{2^n}+1)}{\Gamma(\frac{1}{2^n}-k)}$ gives weird fractions

I stumbled across a series while playing around with a functional equation, it looks like this: $$s(k)=\sum_{n=1}^\infty\frac{\Gamma(\frac{1}{2^n}+1)}{\Gamma(\frac{1}{2^n}-k)}$$ Mathematica gives ...
Loading - 146 Complete's user avatar
0 votes
1 answer
41 views

Show that $E[X] = \frac{\alpha}{\alpha +\beta}$ for the beta probability distribution function

Show that $E[X] = \frac{\alpha}{\alpha +\beta}$ Let X be a continuous random variable we say that $ X ∼ Beta(α, β);\alpha,\beta >0 $ if his density function is: $$f(x) = \frac{1}{B(\alpha,\beta)} ...
samsamradas's user avatar
3 votes
0 answers
61 views

What is the inverse Mellin transform of $\Gamma(1+i s)$?

Mathematica and WolframAlpha both indicate $$ \mathcal{M}_{s}^{-1}\left[\Gamma\left(1 + {\rm i}s\right)\right]\left(x\right) = -\,{\rm i} \operatorname{G}_{\,0,1}^{\,1,0}\,\left(x,{\rm i}\, \left\vert\...
Steven Clark's user avatar
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3 votes
6 answers
246 views

What is a good way to compute $\Gamma(1/3)$ on a standard pocket calculator?

This question is inspired by this one. The earlier question asks how to calculate a certain integral efficiently with a standard pocket calculator. A fine answer by Travis Willse gives a good result ...
Oscar Lanzi's user avatar
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1 vote
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28 views

Numerical Computation of the Gamma Function for large complex numbers

I'm looking for a method to numerically compute the Gamma function $Γ(z)$ for complex numbers of the form $$z= \frac{1}{2} + it,$$ particularly for large values of $t$. Does anyone know of any ...
Felipe Oliveira's user avatar
0 votes
1 answer
42 views

Gamma distribution tricky problem

Suppose that the time, in hours, it takes to repair a pump is a random variable X which has a gamma distribution with parameters α = 2, β = 1/2. what is the probability that a. it takes at most 1 hour ...
samsamradas's user avatar
0 votes
1 answer
51 views

Transformation from squared integral to double integral

for context, I saw a friend of mine did this, is he correct? $$[T(1/2)]^2 = 4(\int_0^{\infty} e^{-x^2}dx)^2$$ which becomes: $$[T(1/2)]^2 = 4(\int_0^{\infty} \int_0^{\infty} e^{-x^2} e^{-y^2}dx dy)$$ ...
samsamradas's user avatar
0 votes
1 answer
40 views

Exponential series with fractional exponents $\sum_{n=0}^{\infty} \frac{x^{\delta n}}{\Gamma(\delta n+1)}$

$$ \mbox{I´m trying to prove that}\quad \sum_{n=0}^{\infty} \frac{x^{\delta n}}{\Gamma(\delta n+1)}\quad\mbox{converges on}\ \mathbb{R}_{\geq 0} $$ For $\delta =1$ we would simpy have $\exp\left(x\...
oli H.'s user avatar
  • 329
4 votes
2 answers
149 views

The sum of $\sum_{k=0}^{n} \binom{n}{k} \Gamma\left(\frac{k}{2}+a\right)$

As the title mentioned, I want to have the value of the sum \begin{equation} \sum_{k=0}^{n} \binom{n}{k} \Gamma\left(\frac{k}{2}+a\right), \end{equation} where $a$ is a positive number, and $\binom{n}{...
Jobs Adam's user avatar
  • 243
0 votes
0 answers
23 views

Derivative of incomplete gamma function w.r.t. the first argument

Wolfram Research provides the following formulas for the derivative of the incomplete gamma function: https://functions.wolfram.com/GammaBetaErf/Gamma2/20/01/01/0002/ https://functions.wolfram.com/...
ufer324's user avatar
  • 25
3 votes
2 answers
114 views

Is it possible to find the $n$th derivative of Gamma function?

By repeatedly differentiating $\Gamma(x)$, I noticed that $$\frac{d^{n}}{{dx}^{n}}\Gamma(x)=\sum_{k=0}^{n-1}\binom{n-1}{k}\psi^{(n-k-1)}(x)\,\frac{d^{k}}{{dx}^{k}}\Gamma(x),$$ where $\psi^{(a)}(x)$ is ...
Ali Shadhar's user avatar
  • 25.8k
2 votes
0 answers
45 views

Proving that $E_{\gamma }\left( z \right) \cdot E_{\gamma }\left( -z \right) \geqslant 1$

I want to prove that $$E_{\gamma }\left( z \right) \cdot E_{\gamma }\left( -z \right) =\sum^{+\infty }_{k=0} \frac{z^{k}}{\Gamma \left( k\gamma +1 \right) } \cdot \sum^{+\infty }_{k=0} \frac{\left( -z ...
ggb's user avatar
  • 21
-1 votes
1 answer
124 views

Why we can find $(-\frac{3}{2})!$ but not $(-1)!$?

I understand that the gamma function is an extension of the factorial function to complex numbers and that it is not defined for non positive integer values. For example, we cannot find the value of $\...
Prince Yadav's user avatar
1 vote
0 answers
39 views

Convergence of integral of gamma function over hankel contour

Let ${H}$ denote hankel contour then I want to show that, $$\oint_{H}t^{z-1}e^{-t}dt$$ Converges for all $z\in\mathbb{C}$. I am familiar that this integral is used in the analytic continuation of ...
RAHUL 's user avatar
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0 votes
0 answers
24 views

Justification of the change of path of integration of $\Gamma$ in this article [duplicate]

I was reading an article and the author wrote \begin{equation*} \int_{z = -2 \pi i \alpha n \mathbb{R}_+}e^{-z}z^{s-1}dz = \int_{z = \mathbb{R}_+}e^{-z}z^{s-1}dz \end{equation*} where $s= 1/2 +...
Gaelink's user avatar
  • 31
1 vote
2 answers
73 views

Evaluate $\int_{0}^1x^n\ln(x)^mdx$ [duplicate]

$$ \mbox{I have to evaluate the integral}\quad \int_{0}^{1}x^{n}\ln^{m}\left(x\right){\rm d}x $$ where I do not have much information about $n,m$, so I suppose they are natural numbers. In that case, ...
MiguelCG's user avatar
  • 345
2 votes
0 answers
36 views

Inequality involving trigamma function $\psi'$

By chance in my ongoing research, I discovered the following inequality, $$a^2\psi'(ax-1) + b^2\psi'(bx-1)\geq 2\psi'(x-1),$$ where $\psi'$ is the trigamma function $\psi'(x) = \frac{\mathrm{d}^2}{\...
Bo Liu's user avatar
  • 181
1 vote
1 answer
95 views

Fourier transform of incomplete gamma function

Ultimately I am interested in the Fourier transform of $$ e^{-i\zeta}(-i\zeta)^{-2\epsilon}\Gamma(2\epsilon,-i\zeta) $$ in a series expansion around $\epsilon=0$, so to first order in $$ \lim_{\...
Tobias's user avatar
  • 133
0 votes
2 answers
101 views

Asymptotic expansion / behaviour of integral function at large x [closed]

How can I find the asymptotic expansion of the following integral, i.e. its behavior for large $x$? $$ \int_0^x \frac{y^k}{\sqrt{1+y^l}}dy $$ I know that the integral can be solved exactly for certain ...
olse barn's user avatar
1 vote
1 answer
36 views

Could there be a function $f$ that satisfies $g(s)=f^{(s)}(x)^s|_{x=1/e}=s!$?

I was thinking about the factorial function today and I wondered: Is there a function $f$ that satisfies $g(s)=f^{(s)}(x)^s|_{x=1/e}=s!$ where the notation is the $s$th derivative of $f$ to the $s$th ...
zeta space's user avatar
0 votes
0 answers
46 views

Bounding $\Gamma(s)\Gamma(1-s) \sin (\pi s)$ to prove Euler reflection formula

I was reading lecture notes of the course $18.785$ offered by MIT. The author makes an error and write $|\sin(\pi s)|$ as $\frac{1}{2}|e^{is} - e^{i\overline{s}}|$ as can be seen in the picture. Could ...
Epsilon-Delta's user avatar
1 vote
0 answers
49 views

Is there a recursive identity for the derivative of the gamma function $\Gamma'(x)$ [closed]

For the normal gamma we have: $$\Gamma(x) = (x-1)\Gamma(x-1)$$ For the digamma we have: $$\Psi(x) = \Psi(x-1) + \frac{1}{x-1}$$ Is there something similar for $\Gamma'(x) = f(x-1)$?
terraregina's user avatar
0 votes
0 answers
58 views

Euler's gamma function and Riemann zeta

On page 7 Edwards gives Euler’s factorial function: $\displaystyle n! = \int_0^{\infty} e^{-x} x^n dx$ for $n=1,2,3,\ldots $ On the next page he gives the same in Gauss’ notation $\displaystyle \Pi(s) ...
zeynel's user avatar
  • 437
5 votes
1 answer
231 views

Closed form for $\Gamma(a-x)$ where $a \in (0,1]$.

Now asked on MO here. I wonder if there is a closed form for $ \Gamma(a-x)$. And by closed form here I mean a finite combinations of elementary functions, powers of $\Gamma(a)$ and powers of $\Gamma(...
pie's user avatar
  • 6,565
0 votes
1 answer
85 views

Proof using the Stirling formula for a limit behaviour

I am trying to understand/prove a lemma which is stated in two papers about branching processes (Lemma 2.2 in "Martingales And Large Deviations For Binary Search Trees" by Jabbour-Hattab and ...
CampFire's user avatar
  • 178
5 votes
2 answers
158 views

$\sum_{n=1}^{\infty} \frac{n 4^n}{(2 n-1)^2(4 n+1)(4 n+3)} \frac{\binom{2 n}{n}}{\binom{4 n}{2 n}}=\frac{4(1+\sqrt{2})}{225}$ [closed]

I want to show that $\sum_{n=1}^{\infty} \frac{n 4^n}{(2 n-1)^2(4 n+1)(4 n+3)} \frac{\binom{2 n}{n}}{\binom{4 n}{2 n}}=\frac{4(1+\sqrt{2})}{225}.$ I tried manipulating the terms in the sum, but that ...
Sam's user avatar
  • 3,360
0 votes
0 answers
43 views

Related to definite integral using equation from table of integrals

I am having the following equation: $P_1 = \int_0^{\infty} \gamma(m_0, \frac{\phi m_0}{\Omega_0 h})\cdot h^{m_1-1} \cdot e^{-\frac{m_1}{\Omega_1}h}\text{d}h$ --- (1) where $\gamma(\cdot,\cdot)$ is ...
Tushar Muratkar's user avatar
4 votes
1 answer
83 views

Showing that $\int_0^1 \frac{\text{d}x}{\zeta(x)\Gamma(x)}<0<1<\int_0^\infty \frac{\text{d}x}{\zeta(x)\Gamma(x)}$ in $3$ minutes, without a calculator

The following question is to be solved within $3$ minutes, without a calculator. $$\text{Let }I=\int_0^\infty \frac{\text{d}x}{\zeta(x)\Gamma(x)}\text{, and let }J=\int_0^1 \frac{\text{d}x}{\zeta(x)\...
Hussain-Alqatari's user avatar
2 votes
0 answers
238 views

Showing $\int_{0}^{1}\frac{E(\tfrac{x}{\sqrt{x^2+8}})}{\sqrt{8-7x^2-x^4}}dx=\frac{1}{3}K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})$

Context $\begin{align} K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1} \end{align}$ and $\begin{align} E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2t}dt\tag{2} \end{align}$ the complete elliptic ...
User's user avatar
  • 323
0 votes
1 answer
85 views

Understanding the proof of Theorem 10.2 in Montgomery & Vaughan's Multiplicative Number Theory

In Theorem 10.2 of the book of Montgomery & Vaughan's Multiplicative Number Theory there are two claims comes without any explanation: 1- For $0 < u < \infty$, $(u + a)^{s−1} ≪ |a|^{σ−1}$ ...
Ali's user avatar
  • 281
0 votes
1 answer
20 views

Once $X\sim \text{gamma}(\alpha = 12, \beta = 2)$ is observed, $Y$ is randomly chosen from $(0,x)$. Evaluate $E(Y)$

Random variable $X\sim \text{gamma}(\alpha = 12, \beta = 2)$. Once $X=x\gt 0$ is observed, $Y$ is randomly chosen from $(0,x)$. Evaluate $E(Y)$. Well I know $E(Y)=E_X[E(Y|X)]$ so I'll proceed by ...
Brenard322's user avatar
1 vote
2 answers
74 views

Asymptotic expansion using gamma functions and Stirling's formula

Question: obtain an expression for the $n$th term of an asymptotic expansion, valid as $ \lambda \to \infty$ for the integral $$I(\lambda) = \int_0^1 t^{2\alpha}e^{-\lambda(t^2+t^3)} dt $$ where $ \...
vegetandy's user avatar
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