Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions.

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2
votes
2answers
36 views

Generalizing Legendre's Theorem on prime factorization of factorials

From Legendre's Theorem we know $$n!=\prod_{p}p^{\lceil \frac{n}{p}\rceil +\lceil \frac{n}{p^{2}}\rceil +.. }$$. Since $\Gamma (n+1)=n!$, i wonder if there is a generalization of this formula for ...
0
votes
0answers
11 views

Gamma Duplication formula with multiplicative constant

i know the duplication formula $\prod_{k=0}^{n-1} \Gamma(z+\frac{k}{n}) =\Gamma(z)\Gamma(z+\frac{1}{n}) \Gamma(z+\frac{2}{n})...\Gamma(z+ \frac{n-1}{n}) = ...
0
votes
0answers
16 views

Solving summation with incomplete gamma function

I solved an indefinite integral that gave me the following result: \begin{equation} f(u) = -\sum_{k=0}^{n}\binom{n}{k}\frac{\mathrm{sgn}(u)^{k+1}\, b^{n-k}}{2\, a^{\frac{k+1}{2}}}\, ...
2
votes
3answers
52 views

Can a complex argument into this function ever yield a real result?

I have a function defined as: $f(z)=\frac{\Gamma{(z)}+1}{z}$ Are there any $z ∈ C$ (with nonzero imaginary part) such that $f(z)∈R$? I tried substituting in $z=a+bi$ with $b≠0$ into the product ...
0
votes
1answer
17 views

Solving integration equation with binomial expression using gamma function [closed]

Can anyone help me how they have solved this? Denote $\beta_1 = (\alpha-1)/2$ and $\beta_2 = (\alpha+1)/2$. The total power received at the BS is $$P_{bs}^{tot} = \int_{-\infty}^\infty ...
2
votes
1answer
51 views

Euler gamma function differential

Do you have any piece of advice on how to calculate a differential of an Euler Gamma function for $ x \in R- Z $?Thank you for all your answers.
1
vote
1answer
24 views

About Beta function $B(\alpha,r\alpha +1)‎\rightarrow‎ 0$

I want to show that $$B(\alpha,r\alpha +1)‎\rightarrow‎ 0$$ when $r‎\rightarrow‎ \infty$ and $0< \alpha <1$. with thanks
2
votes
1answer
60 views

Integral with incomplete gamma function

I am trying to solve this integral: \begin{equation} \frac{1}{c^{b}}\int_{0}^{\infty} x^{n}\, e^{-a x}\, \gamma(b,c(-d+x)) \ \mathrm{d}x \end{equation} where, $n>0$ is an integer, and $a$, $b$, ...
-2
votes
0answers
22 views

integral with two lower incomplete gamma

Can I get a step by step answer to this integration ? $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$
0
votes
0answers
29 views

Integral involving $\Gamma$ function

I am in the middle of a proof and I find myself confused on one step. I believe there is some sort of algebraic manipulation or substitution that I am perhaps missing. $$ \sum _{m = 0} ^ \infty ...
0
votes
1answer
40 views

integral include lower incomplete gamma

I am trying to calculate the following integral: $$ \int_0^{\infty}e^{-\beta x}\gamma(\alpha,\theta x)dx $$ where all parameters are positive. Any help , Thanks!
-2
votes
0answers
40 views

Prove using gamma function

prove $$\Gamma\left(\frac{1}{n}\right)\Gamma\left(\frac{2}{n}\right)\Gamma\left(\frac{3}{n}\right)\Gamma\left(\frac{4}{n}\right)\cdots ...
1
vote
1answer
57 views

Showing that $\int_{\mathbb{R}} e^{x(t+1)} \left(1 + \frac{e^x}{k} \right)^{-(k+1)}dx = k^{t+1} \frac{\Gamma(k-t)\Gamma(1+t)}{\Gamma{(k+1)}}$

I want to show that $$ f(t) = \int_{\mathbb{R}} e^{x(t+1)} \left(1 + \frac{e^x}{k} \right)^{-(k+1)}dx = k^{t+1} \frac{\Gamma(k-t)\Gamma(1+t)}{\Gamma{(k+1)}}$$ Here, $k,t \in \mathbb{R}$. I have ...
0
votes
0answers
31 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to ...
1
vote
0answers
105 views

Integral involving the gamma function

If we define $$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \cos(2\pi{x})}$$ how would one go about evaluating $$ \int_1^R \frac{1}{x} \log{f(xe^{i\alpha})} dx$$ for some parameter ...
1
vote
0answers
108 views

If $f_1(k)$ and $f_2(k)$ reach their max. at $k_{m1}$ and $k_{m2}$, resp., show that $k_{m1} > k_{m2}$ in the following case

Let $k$ represent an integer value. We define function: $f_1(k)=k(1-k t)\frac{\Gamma(L-k,k a_1)}{\Gamma(L-k)}$, for $1\le k\le L-1$, with $kt \le 1$, and $a_1$, $t (<1)$ are some positive ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...
1
vote
1answer
30 views

(Ahlfors, p198) Why is it clear we can write $G(z-1)=ze^{\gamma(z)}G(z)$ when deriving the Gamma function?

In Complex Analysis by Ahlfors (p198), the author starts with the functional $$ G(z) = \prod_1^\infty \left( 1 + \frac{z}{n} \right) e^{-z/n} $$ and goes on to state that we may obviously write $$ ...
1
vote
3answers
50 views

How could I solve $\int_{-\infty}^{+\infty} x^2e^{-x^2}dx$ apply special function gamma

I try solve the integral $$\int_{-\infty}^{+\infty} t^2e^{-t^2}dt$$ I do not know but I think that I should apply $gamma\ function$, which is $$ \Gamma (x)=\int_{0}^{\infty} t^{x-1}e^{-t}dt$$ Like ...
5
votes
2answers
147 views

How could one solve $\int_{0}^{\infty} \frac{1}{1-t^4}dt$ with special functions?

How could one solve $$\int_0^\infty \frac{1}{1-t^4} \, dt\,?$$ I have to apply special functions, so I thought that I have to use the change variable $$u=t^4,$$ but $$du=4t^3\,dt$$ and when ...
1
vote
1answer
106 views

Gamma function and sine

I need to prove that the function $$c(\alpha)=\sin\left(\frac{\pi \alpha}{2}\right)\cdot \Gamma(\alpha)$$ is decreasing on $(1,2]$. It is evident if you build the graph of it. E.g. in WolframAlpha you ...
4
votes
1answer
49 views

Show that the gamma function converges

One way to define the gamma function is $$\Gamma(x)= \lim_{p\rightarrow \infty} \Gamma_p(x) $$ Where $$\Gamma_p(x)=\dfrac{p! p^x }{x(x+1)...(x+p)}=\dfrac{p^x}{x(1+x/1)(1+x/2)...(1+x/p)} $$ How to ...
1
vote
1answer
43 views

Proof that $ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4 $?

Can one prove that $$ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3),\text{ for }2<\Re(d)<4? $$ I would prefer using the methods of contour integration.
0
votes
1answer
40 views

Asymptotic expansion question

How may I use Watson's Lemma to find the full asymptotic expansion for; $$ I(\lambda)= \int_0^\infty e^{-\lambda(1+s)}ln(1+s^2)ds $$ as $\lambda \rightarrow \infty$. Thanks in advance
7
votes
1answer
129 views

Why does this double infinite sum converge to $e$?

I can't seem to come to grips with the result below: $$S=\sum_{n=1}^\infty \sum_{k=n}^\infty\frac{1}{k!}=e$$ which is given by Mathematica (code below) and (numerically) verified by WolframAlpha. ...
0
votes
0answers
18 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
0
votes
0answers
31 views

Unclear passage in integration involving Gamma functions

I find myself in need of some advice on an integration problem. Let $F(x,\lambda)=\Gamma(x,\lambda)/\Gamma(x)$, $x,\lambda>0$ be the Regularized Upper Incomplete Gamma Function, where ...
0
votes
0answers
22 views

A multiple of the Gamma Function when integrated between $0$ and $\infty$

I was reading through this answer on stats.stackexchange, but didn't follow the mathematics behind one step. They have $$\int\limits_{\tau=0}^{\infty} e^{-\tau( ...
-1
votes
0answers
47 views

How to evaluate integral $\int_1^n \frac{A^x}{\Gamma(x)}\mathrm{d}x$?

How to evaluate the following integral ? $$\int_1^n \frac{A^x}{\Gamma(x)}\mathrm{d}x,\quad A>1,n\geq 2,n\in\mathbb{N}$$ The Gamma function in the denominator seems bothersome. Thanks- mike
1
vote
0answers
27 views

Derivative of the upper incomplete gamma function.

I wish to compute the derivative of the upper incomplete gamma function \begin{equation} \Gamma(s,x) = \int_{x}^\infty t^{s-1}e^{-t} \, dt . \end{equation} Wikipedia states of the derivative of ...
1
vote
2answers
46 views

A simple inequality, does it always hold?

I am looking at the following inequality: $$\Gamma(-\frac{x+1}{x})\lt x , \forall x\gt \frac{7}{2}$$ It seems that the LHS and RHS eventually diverge for large enough $x$, but I have failed in a ...
4
votes
1answer
125 views

A $\log \Gamma $ identity: Where does it come from?

$$\log \Gamma (n)=n\log n -n +\frac{1}{2} \log \frac{2\pi}{n}+\int_0^\infty \frac{2\arctan (\frac{x}{n})}{e^{2\pi x}-1} \,\mathrm{d}x$$ Is an identity that is derived from using Sterling's ...
2
votes
0answers
70 views

Closed Form for the Integral: $\int_0^1 t^n\log\Gamma(t+a)dt$

I am wondering if someone could tell me whether or not the following integral has a closed form representation: $$\int_0^1 t^n\log\Gamma(t+a)dt$$ In Srivastava's and Choi's wonderful book Zeta and ...
0
votes
1answer
25 views

Calculate the Gamma function Γ(2.7) [duplicate]

\begin{align} \displaystyle Γ(2.7) &= \int_{0}^{∞} x^{(2.7-1)} \cdot e^{(-x)} dx \\[7pt] Γ(2.7) &= 1.54968 \end{align} I got this answer online, but I do not know how to ...
1
vote
1answer
21 views

Unit Measure Axiom for the Gamma Distribution

I'm studying basic probability and in my lecture notes, it shows how the Gamma function results from the convolution of two exponential random variables. To introduce the gamma function it shows that ...
1
vote
0answers
25 views

Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
10
votes
1answer
3k views

Closed form for an integral involving the incomplete Gamma function?

Let $\alpha>1$, $K>1$ and $n \in \mathbb{N}^+$, looking for a closed form solution to a tough integral $$I(\alpha,K,n)=-\int_{-\infty }^{\infty } \frac{2 i e^{-i K u} (K u-i) \Bigl[\alpha \, (-i ...
1
vote
0answers
56 views

Is solution to $\Gamma(x+1)=121$ algebraic?

If I had the following: $$x!=\Gamma(x+1)=121$$We see that $x\approx5$. But is the exact value of $x$ algebraic? For some non-whole number $x$ input that is algebraic, I think the output of the Gamma ...
2
votes
1answer
53 views

Which theorems in the Gamma Function are important? [closed]

I'm interested in the Special Functions especially the Gamma Function. I decided to write a Bachelor Thesis about it but I do not know what kind of theorem(s) in the Gamma Function that is (are) very ...
5
votes
2answers
80 views

Improper integral of $\log x \operatorname{sech} x$

How to prove the following? $$ \int_0^\infty \log x \operatorname{sech}x\,dx = \frac{\pi}{2} \log\left( \frac{4\pi^3}{\Gamma(1/4)^4} \right) $$ I obtained the right side with CAS. It seems like this ...
0
votes
2answers
57 views

Deduce that $\Gamma(x)=\lim_{n\to\infty}\frac{n!x^n}{x(x+1)\cdots(x+n)}$ for all $x$, given that it holds for $0<x<1$

How to understand rigorously that $(95)$ hold for any $x>0$ using that $\Gamma(x+1)=x\Gamma(x)$. Can anyone explain please this strictly?
0
votes
0answers
62 views

Asymptotics relationship from algebric identity?

Background We start with the identity: $$ \sum_{r=1}^n r \ln r + \ln (r-1)! = n\ln n! $$ $$ \implies \sum_{r=1}^n \frac{r}{n} \ln r + \frac{1}{n} \ln\Gamma(r) = \ln n!$$ $$ \implies ...
4
votes
2answers
89 views

Calculate $‎\lim‎_{ ‎r\rightarrow ‎\infty‎}‎‎\frac{\Gamma(r\alpha)}{\Gamma((r+1)\alpha)}‎‎$

I need to calculate limit $$‎\lim‎_{ ‎r\rightarrow ‎\infty‎}‎‎\frac{\Gamma(r\alpha)}{\Gamma((r+1)\alpha)}‎‎$$ where $0<\alpha <1$ and $\Gamma(.)$ is Gamma function. with thanks in advance.
0
votes
0answers
26 views

Gamma function converges to zero

I want to show that for $x \in [1,2]$ the Gamma function $\Gamma(x+iy)$ converges uniformly to zero as $y \rightarrow \pm \infty.$ Unfortunately I have not found a suitable representation of the Gamma ...
0
votes
0answers
11 views

Moment relations of certain probability distributions

My question is concerned with moment relation of a certain subclass of probability distributions. Concretely, if $X$ is Normal-distributed with mean $0$ and variance $1$, it is well known that the ...
1
vote
1answer
42 views

(Ab)using the factorial and gamma functions

I have a product of the following form: $$ P = (N-\alpha)(N-2\alpha)\cdots(N-k\alpha) $$ where $k$ is an integer between $1$ and $N-1$, and $\alpha$ is a real number in $(0,1]$. Clearly, for ...
4
votes
2answers
97 views

$\frac{\Gamma(\frac{n_T + n_C - 2}{2})}{\sqrt{\frac{n_T+n_C-2}{2}}\Gamma\left(\frac{n_T+n_C-3}{2}\right)} \approx 1 - \frac{3}{4(n_T+n_C+2)-1}$

In this answer to a question I asked (which derives the variance of Cohen's $d$), the approximation $$\frac{\Gamma\left(\frac{n_T + n_C - ...
0
votes
0answers
19 views

An asymptotic behaviour of gamma functions [duplicate]

Let $x \in \mathbb{R}$, and consider the limit of the ratio \begin{equation} \lim_{x \to +\infty} \frac{\Gamma(x,x)}{\Gamma({x})}, \end{equation} where $\Gamma(x,x)$ is the upper incomplete Gamma ...
0
votes
1answer
23 views

Find the Laplace transform of the Gamma pdf

Per wikipedia the Laplace transform of the gamma distribution is $$L_X(s) = (1+\theta s)^{-k} = \frac{\beta^\alpha}{(s+\beta)^\alpha}$$ As an exercise I would like to show this.The definition I have ...
1
vote
0answers
54 views

Is it possible to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial zero of Riemann zeta function?

I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty ...