Questions tagged [gamma-function]
used only for functions based on gamma, not functions with some obscure relation to gamma
165
questions
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Exponential-like function equivalent for the Dixonian Elliptics
Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?
2
votes
1
answer
276
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Stirling's formula and a Gamma function relation
I am trying to understand a paper by by A. Booker on poles of Artin $L$-functions where in one of the lemmas he uses the following identity, derived using Stirling's formula:
$$
\frac{\Gamma(s/2)^2}{2^...
8
votes
1
answer
1k
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The inverse of the digamma function
The gamma function is increases on the interval $(x_0, \infty),$ where $x_0$ denotes the unique zero of the digamma function on the positive half line.
The inverse function of gamma function defined ...
5
votes
1
answer
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How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?
I tried to find an inverse Laplace transform by series as follows
$$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$
and by ...
0
votes
1
answer
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Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$
I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$
I've tried Mathematica, but it does not converge to a solution....
58
votes
8
answers
35k
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Inverse gamma function?
This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this.
We have the ...
2
votes
1
answer
327
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Upper bound for the complex Beta function
The question is almost the same as here.
What is the upper bound for a complex Beta function $\displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)}$ with $0<Re(s)<1$ and $0<Re(z)<1$;...
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1
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How to prove the convexity of a simple function involving a ratio of two polygamma functions?
Let
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0
\end{equation*}
and
$$
\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.
$$
In the literature, ...
18
votes
2
answers
1k
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Formula for volume of $n$-ball for negative $n$
Does the expression $$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n,$$ which gives the volume of an $n$-dimensional ball of radius $R$ when $n$ is a nonnegative integer, have any known ...
0
votes
1
answer
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Is $\Gamma(z,1)\not=0$ for all $z$ with $\Re(z)<0$?
I found this paper online which appears to present zeros of the incomplete gamma function within the right half plane. It makes me think that there are no zeros in the left half plane. Not sure how to ...
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T functions arising from derivatives of incomplete Gamma function
Here the derivatives of the incomplete gamma functions are described via:
$$
T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right|...
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6
answers
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One-line proof of the Euler's reflection formula
A popular method of proving the formula is to use the infinite product representation of the gamma function. See ProofWiki for example.
However, I'm interested in down-to-earth proof; e.g. using the ...
1
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1
answer
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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
This is a repost from MSE because I got no answers there.
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 ...
1
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0
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114
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Question on Artin's Gamma function on $\operatorname{SO}(2,0)(\mathbb R)$
$\DeclareMathOperator\SO{SO}$Let $G=\SO(2,0)(\mathbb{R})$, a quasi-split group with signature $(2,0)$. Let $e$ be an element in $O(2,0)(\mathbb{R}) \setminus \SO(2,0)(\mathbb{R})$.
Let $\pi$ be an ...
5
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3
answers
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The exact constant in a bound on ratios of Gamma functions
The answer to another question (Upper bound of the fraction of Gamma functions) gave an asymptotic upper bound for an expression with Gamma functions:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
11
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1
answer
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New method to compute square roots [closed]
In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
2
votes
2
answers
571
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Integral calculus with Gamma function [closed]
I have to prove that for $0<\alpha<1$ and $\beta>0$,
\begin{equation}
\int_{0}^{\infty} x^{-\alpha-1}\left(e^{-\beta x}-1\right)dx=\beta^\alpha\Gamma(-\alpha),
\end{equation}
and I have ...
0
votes
1
answer
175
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Infinite limit of sums of gamma functions is constant?
The following expression arises in the study of hierarchical models. I suspect that the sum of the underlined $4$ terms become constant as $\alpha\rightarrow \infty$. Mathematica agrees when prompted ...
3
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0
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Derivation of an integral containing the complete elliptic integral of the first kind
I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5).
$$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{...
3
votes
4
answers
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Some Log integrals related to Gamma value
Two years ago I evaluated some integrals related to $\Gamma(1/4)$.
First example:
$$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{...
5
votes
3
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Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
0
votes
0
answers
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Incomplete Gamma function $\Gamma(0,x)$ and $\Gamma(0,-x)$
I want to find the value of this
\begin{align}
y=\Gamma(0,x)-\Gamma(0,-x)
\end{align}
where $\Gamma$ is the upper incomplete Gamma function, $x>0$ is real. I can't find the definition of $\Gamma(0,-...
-1
votes
1
answer
177
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Conjecture about the equality : $f(y)=y\ln(y)+\sum_{n=2}^{\infty}\pm\frac{\left(y\ln y\right)^n}{2^{a_n}}$
I try here because I expect I cannot have any answer on MSE :
Problem :
Let :
$$f\left(x\right)=\frac{\left(1+\ln27\right)x!\ln x!}{x+1},g\left(x\right)=x\ln x$$
Then it seems $\exists y\in(0,1)$ and $...
1
vote
0
answers
143
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polynomial approximation of hypergeometric function 2F1
I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations:
$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
0
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0
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An interpolation of $n!$ such that its derivatives have few zeros
The $\Gamma$-function restricted to $(0,+\infty)$ has the following properties:
$\Gamma(n)=(n-1)!$ for $n=1,2,3,...$.
The $k$'th derivative $\Gamma^{(k)}$ has no zeros on $(0,+\infty)$ when $k$ is ...
5
votes
0
answers
414
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Determinant of Hankel matrix with $a_n=(n!)^2$
Consider a Hankel matrix of the form
$H_n(a_0(n))=\begin{pmatrix}
a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\
(1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\
(2!)^2 &...
1
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0
answers
67
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Uniform bound on product of Gamma functions in an article by Jerison and Kenig
I am new here. I'm reposting a question I originally posted here on Math Stack Exchange. I realized that maybe this is more appropriate place to ask such a question...
I have been trying to read ...
0
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0
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140
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Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?
In order to see what happens when taking the functional equation in this form:
$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\...
22
votes
4
answers
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Who invented the gamma function?
Who was the first person who solved the problem of extending the factorial to non-integer arguments?
Detlef Gronau writes [1]: "As a matter of fact, it was Daniel Bernoulli who gave in 1729 the ...
5
votes
2
answers
244
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Inverse Mellin transform of 3 gamma functions product
I want to calculate the inverse Mellin transform of products of 3 gamma functions.
$$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$
Above contour integral has ...
1
vote
2
answers
550
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Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
2
votes
0
answers
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Motivation behind the Bohr-Mollerup Theorem relating the Gamma function
In Wikipedia, it states about the Bohr-Mollerup Theorem:
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
If anyone knows, ...
1
vote
1
answer
482
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Upper bound of the fraction of Gamma functions
Is there a simple upper bound of the following fraction of Gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...
5
votes
0
answers
645
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Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
5
votes
2
answers
345
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Extended binomial coefficients and the gamma function
For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (...
0
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0
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Addition formulas for q-analogs of trigonometric functions
Sine and Cosine functions possess notable formulas for addition of angles
$$
\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b).
$$
One can ...
2
votes
1
answer
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Gamma function and the somewhat extended version of Bohr-Mollerup theorem
The Gamma function $\Gamma$ is defined by
\begin{equation*}
\Gamma(x)=\int_{0}^\infty t^{x-1}e^{-t} \,\mathrm{d}t,
\end{equation*}
for $x>0$. It satisfies the well-known functional equation
$$\...
5
votes
0
answers
610
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The Basel problem revisited?
In the Basel problem, the $sinc$ function is considered at the Wikipedia page.
Let me try to make an alternative function definition:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
1
vote
1
answer
101
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Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function
Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series
$$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
3
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2
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$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function
I try to calculate the following series
\begin{align*}
S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k},
\end{...
8
votes
3
answers
2k
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Multiplicative integral of $\Gamma(x)$
A recent question on the notion and notation of multiplicative integrals
( What is the standard notation for a multiplicative integral? ) induced me to play with the Riemann products of the Gamma ...
1
vote
1
answer
121
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Sum of reciprocal of Pochhamer symbols through multiples of a natural L
In a question in StackExchange (https://math.stackexchange.com/questions/4236635/sum-of-quotients-of-gamma-functions), I asked if there is a closed expression for the following sum related with a ...
4
votes
2
answers
674
views
Generalized binomial coefficients and Gaussian density
I ran into an expression calculating the expected value of $\exp(i t \sigma)$ where $\sigma$ is the total number of cycles in a uniformly chosen $S_n$ element. The expression is
$$E_n (\exp(i t \sigma)...
0
votes
1
answer
232
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Logarithm of an integral involving generalized real binomial coefficients
I could not find a closed form for this integral although I think it should have been studied.
What is a good approximation to $I$ in
$$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...
3
votes
0
answers
394
views
Generalization of Trigonometric, Hyperbolic, $\Gamma$ and $\zeta$ Functions
It is becoming increasingly clear that the expression $~\displaystyle\prod_{n\in\mathbf M}\left[1-\frac{z^s}{(n-a)^s}\right],~$ with $~|\mathbf M|=\aleph_0,~$
$a\not\in\mathbf M,~$ and $~\Re(s)>1,~$...
4
votes
1
answer
445
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Beta function, harmonic numbers, and integral values
A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k
$$
where $\beta_x( -1 - ...
4
votes
2
answers
195
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Hankel determinant of incomplete gamma functions
I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the ($r \times r$ form)
I'd like to evaluate these determinants. Elementary operations help, but ...
5
votes
1
answer
708
views
On the integral $\int_0^1\log(x!)dx$ revisited
I was interested in an integral that I known from [1], it is
$$\int_0^1 \log(x!)dx.$$
I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram ...
3
votes
0
answers
229
views
definite integral with incomplete gamma function and exponential
While working with electron density computations in quantum chemistry, I encountered the following improper integral:
$$
I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
1
vote
1
answer
182
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Good upper bound for $\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$, where $a,b \in (0, 1)$ and $N \ge 1$
Let $a,b \in (0, 1)$ and $N \ge 1$, and consider the incomplete gamma function $x \mapsto \Gamma(1-a,x)$.
Question
Is there a simple bound (involving 'simple function's) for the expression $\Gamma(1-...