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

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Can the hypergeometric function be extended analytically to the complex plane in the interval [1,$\infty$ )?

Just a thought. The hypergeometric function, which can be written as: $$F(a,b,c \space;z) = \frac{\Gamma (c)}{\Gamma (b) \Gamma (c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt$$ is obviously ...
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2answers
94 views

Ahlfors “Prove the formula of Gauss”

He says: Prove the formula of Gauss: $$ (2\pi)^\frac{n-1}{2} \Gamma(z) = n^{z - \frac{1}{2}}\Gamma(z/n)\Gamma(\frac{z+1}{n})\cdots\Gamma(\frac{z+n-1}{n}) $$ This is an exercise out of Ahlfors. ...
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1answer
27 views

complex integral of z to the power alpha

I would like to perform an inverse laplace and at some point of the calculation I have to compute this integral $$\int_{\gamma-i\infty}^{\gamma+i\infty} z^{(1+n)\alpha-1}e^{z} \frac{dz}{2\pi i}$$ ...
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1answer
45 views

The finite sum $\sum_{k=0}^{n} \frac{1}{k!(n-k)!\Gamma(k+\frac{1}{2}) \Gamma(a-k+ \frac{1}{2})} $

I ran across a few finite sums involving the gamma function while doing some research about special cases of the hypergeometric function. They're all very similar, so I'll just post one of them. ...
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1answer
167 views

Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$

I came across this nice identity: $$\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$$ Is there an elementary proof?
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1answer
71 views

Gamma Function Contour Integration

So, I've been trying to prove the following integral related to the gamma function, and I'm really banging my head against the wall over this: ...
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22 views

Question concerning the gamma function in relation to other holomorphic functions when $Re(\xi) > 0$

Let $f$ be indefinitely differentiable on $\mathbb R$ that has compact support. $\implies f$ belongs to the Schwartz space. Consider: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
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0answers
68 views

integral involving incomplete gamma function

Need to evaluate the integral $$ \int_a^b e^{1/x}\,\Gamma(m,1/x)\,dx $$ or equivalently $$ \int_{1/a}^{1/b} y^{-2}\,e^{y}\,\Gamma(m,y)\,dy, $$ where $m$ is an integer, and $0<a<b<\infty$. The ...
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32 views

For which values does the Gamma function yield an integer result?

Gamma Function: $$\Gamma(t)=\int_{0}^{\infty}x^{t-1}e^{-x}dx$$ Is it known for which values of $t$ (real or complex), the value of $\Gamma(t)$ is integer? Are there any known specific patterns of ...
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1answer
34 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct the following probabilistic model, \begin{equation} \begin{split} \frac{1}{\mu}\int^{\infty}_{0}P(N \geq n\, |\, L=l, T=t)\,e^{-\frac{l}{\mu}} dl &= ...
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0answers
31 views

Calculating $\Gamma(2n+1)$

I know that $$ \Gamma(x+1) = x*\Gamma(x) $$ $$ \Gamma \left (\frac{1}{2} \right) = \sqrt{π} $$ $$ \Gamma(n) = (n-1)! $$ $$ \Gamma(n+ \frac{1}{2}) = \frac{1*3*5* ... (2n-1)}{2^{n}}*\sqrt{π} $$ But ...
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0answers
13 views

Relationship between gamma and inverse gamma distributions under some algebraic operations

I have a question about the relationship between gamma and inverse gamma distributions. I have an equation that takes $L/(c*X)$ Where $X$ is a Gamma distribution and $c$ and $L$ are constants I'd ...
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1answer
23 views

Proving that $\Gamma (x) = \int_{0}^{1} \left( (ln \left(\frac{1}{u} \right) \right)^{x-1} du$

I want to prove that $$ \Gamma (x) = \int_{0}^{1} \left( (ln \left(\frac{1}{u} \right) \right)^{x-1} du $$ I start with $$ \int_{0}^{1} \left( ln \left(\frac{1}{u} \right) \right)^{x-1} du = $$ ...
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1answer
48 views

Gamma like integral with power in exponent

I came across an integral in my stat mech book of the form $$\int_0^\infty x^{d-1}e^{x^s}dx$$ The book claims without proof that this is $$\frac 1s\Gamma(d/s)$$ I tried doing a change of variables ...
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2answers
83 views

Taylor Series looks like exponential

guys! I need a little help. I have this series $$ \sum_{k \ge 0} \frac{\Gamma(j)}{\Gamma(j+k/2)}(-t)^k $$ where $j \in \mathbb{N}$. I need to know if the limit of this function when $t$ goes to ...
2
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0answers
36 views

When this inequality true?

If $a$ and $b$ are non-negative integers and $c$ and $d$ are non-negative real numbers, for what values is the following inequality true? $\log((a+b)!) - \log(a!b!) \ge(a+b) \log(c+d) - (a \log(c) ...
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0answers
13 views

Gamma function approximation

I was told that for Gamma function to get the following approximation $\Gamma(2f(x))\approx \log(x)$ for large $x$, function $f(x)$ has to behave like $\log(\log(x))$. I need to use it, but first ...
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0answers
18 views

Bound of LogGamma and Log

It's true this inequality: $\log(\Gamma(a+1) - a * \log(b) > (\log(\Gamma(a1+1) - a1 * \log(b1)) + (\log(\Gamma(a2+1) - a2 * \log(b2))$ Where: $a = a1 + a2$ $b = b1 + b2$ and $a,b,a1,a2,b1,b2 ...
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2answers
44 views

Integration by using special functions

$$\int ^{\pi }_{0}\dfrac {dt}{\sqrt {3-\cos t}}$$ How can you solve the following equation by using alpha/gamma functions and putting $$\cos t=1-2\sqrt {u}$$
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3answers
83 views

Expressing $\prod_{k=1}^n \left( k - \frac{1}{2} \right)$ using the gamma function

I want to express $$\prod_{k=1}^n \left( k - \frac{1}{2} \right)$$ using the gamma function. I think this is equivalent to $\left(k-\frac{1}{2}\right)!$ so I set $a=k-1$ and then used the identity ...
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1answer
22 views

Upper incomplete gamma integral

I would like to know whether the following relation is correct or not? $\frac{d}{dz}\Gamma(w,\mu z)= -\mu^wz^{w-1}e^{-\mu z}$, where $\Gamma(w,\mu z)$ is the upper incomplete gamma integral. Can ...
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2answers
139 views

Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.

$$ \int_{-\pi/2}^{\pi/2}\cos^m\left(x\right){\rm e}^{{\rm i}n x}\,{\rm d}x ={\pi \over 2^{m}}\, {\Gamma\left(1 + m\right) \over \Gamma\left( 1 + \left[m + n\right]/2\right)\ \Gamma\left( 1 + ...
2
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2answers
94 views

Integral calculus question relating to particle motion

"A particle of mass m is attracted toward a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at distance L, from 0, find ...
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0answers
20 views

identity involving gamma function

Is this identity $$ \lim_{x\to0}[\Gamma(x+\imath b_1)+\Gamma(x-\imath b_2)]=2\pi\delta(b_1-b_2),\;\imath=\sqrt{-1}. $$ supposed to be known? Any reference?
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1answer
99 views

Integral of incomplete gamma function

I am trying to integrate this: $$\int_0^\infty z^{-|M|-1}\,\Gamma(A,z)\;dz$$ where $A$ is a real positive, and note that the power of $z$ is $-|M|-1$, i.e., is forced to be negative real.
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1answer
35 views

Approximation of a factorial

In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which ...
0
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1answer
22 views

How to derive the Dirichlet-multinomial?

Common knowledge The multinomial: $$p(z|\theta) = \frac{n!}{\prod_i z_i!} \prod_i \theta_i^{z_i}$$ And the Dirichlet: $$p(\theta|\alpha) = \frac{1}{B(\alpha)}\prod_i \theta_i^{\alpha_i-1}$$ with ...
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1answer
51 views

Expansion of lower incomplete gamma function $\gamma(s,x)$ for $s < 0$.

The lower incomplete gamma function for positive $s$ is defined by the integral $$ \gamma(s,x)=\int_0^{x} t^{s-1} e^{-t} dt. $$ Taylor expansion of the exponential function and term by term ...
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56 views

Learning about the gamma function.

I have just started learning about the gamma function but the books I have are not sufficient to give me a complete picture of it. Can you guys suggest some online resources/free books where I can ...
8
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1answer
181 views

Strange closed forms for hypergeometric functions

So in the process of trying to find a derivation for this answer, the following interesting equalities arose (one can check with Wolfram Alpha/Mathematica): $$\frac{8\sqrt{2}G^4}{5\pi^2} ...
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0answers
54 views

relationship of gamma and beta functions

I was reading a article on proving the relationship between gamma and beta functions but there are some things I don't understand. Then the variables are changed as Then they have obtained ...
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3answers
37 views

domain of gamma function

When gamma function is defined $\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}dt.$ for s>0.But then it says that " The gamma function is defined for all complex numbers except the negative integers and zero. " ...
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0answers
33 views

Prove an equation about fractional integral

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Can anyone help me solve this? I really appreciate. If $f$ is continuous on $[0, \infty)$, for $\alpha \gt ...
4
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1answer
63 views

Why $dt/t$ in Mellin transform

I've noticed that often when people write the Gamma function $\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt$, that they write it like $$ \Gamma(s) = \int_0^\infty t^s e^{-t}\,\frac{dt}{t} , $$ where ...
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0answers
33 views

What does the $z_0$ and $\mathcal{O}$ mean in the following function?

I am trying to figure out what the subscript next to the 'z' represents at the following formula: $$\frac{1}{\Gamma(z)}=\frac{1}{\Gamma(z_0)}(1+\mathcal{O}(z-z_0))$$ Thanks!
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1answer
21 views

Gamma: Find P[15<x<20] for X~GAM(5,4)

I'm having a lot of trouble with this Gamma distribution problem X~GAM(θ,k). My text book is terrible and doesn't have any examples like this: Find P[15 < X <20] for X~GAM(5,4) Please note my ...
0
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1answer
32 views

theory of Gamma function

in the integral definition of the gamma function $$\Gamma(x)=\int_0^{\infty}e^{-t}t^{x-1}dt$$ in order to find the gamma function as an infinite limit, we use the following lemma, ...
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2answers
44 views

Expressing $\mathrm{B}(\sinh(x), \cosh(x))$ in terms of elementary functions

Is it possible to express: $\mathrm{B}(\sinh(x), \cosh(x))$ (where $\mathrm{B}$ is the beta function) In closed form, in terms of elementary functions?
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1answer
65 views

Why $x$ should be **real**

Show that if $x$ is real and positive ($x \in \mathbb{R}^{> 0}$) then: $$Γ(x)= ∫_0^1(- \ln ⁡t )^{x-1} dt $$ Solution : First recall the gamma function: $$Γ(x)= ∫_0^∞ e^{-t} t^{x-1} dt \; ...
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3answers
101 views

Is there a relatively elementary proof that the gamma function has no zero?

I have never studied functional analysis. So I am unable to understand terms such as meromorphic, holomorphic and etc. So far, I have showed Gauss', Euler's , Weierstrass' definition of the Gamma ...
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0answers
44 views

Analytic continuation of the $\Gamma$-function

I'm currently working from Zwiebach's A First Course in String Theory, Second Edition, and I am wondering about the following question (which is paraphrased): "Use the equation $$ \Gamma(z) = ...
0
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2answers
46 views

Other representation of Gamma Incomplete function

I have a question regarding Gamma Incomplete function: In the "Table of Integrals, Series, and Products, Seventh Edition" equation 8.353.3 page 900, there is a defenition for the incomplete gamma ...
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0answers
23 views

How do i prove a given function is measurable?

Lebesgue Integral - Frank Jones p.201 I'm trying to prove $\Gamma(z)\Gamma(w)=\Gamma(z+w)B(z,w)$ ($z,w$ are complex numbers with positive real part). If you know an easier way to prove this, please ...
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1answer
22 views

Limit of a binomial sequence is Poisson? [closed]

Let X be a sequence of random variables with binomial distribution $\mathcal{B} (n, \lambda /n)$, where $\lambda > 0$. Show that $$ \lim_{n \to \infty} \mathbb{P}[X_n = k] = e^{-\lambda} ...
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0answers
69 views

How to prove this inequality about digamma function?

Let $\psi$ be the digamma function, such that, $\psi(x) = \Gamma'(x)/\Gamma(x)$. How can I show that $\log x - 1/x < \psi(x) < \log x - 1/(2x)$.
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1answer
35 views

Multiplying two Gamma distributions over the same variable

I am looking at a software library where there is a function that multiplies two Gamma distributions defined over the same random variable. So, it is basically multiplying two Gamma densities with ...
1
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1answer
17 views

multiplying gamma function over different input variables

I have a function (the Gamma function) defined as: $$ F(\alpha) = \int_{0}^{\infty} x^{\alpha-1} e^{-x}dx $$ Now, I want to multiply this function evaluated at two points as: $$ F(\alpha_0) = ...
0
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1answer
60 views

Asymptotes of $\Gamma(\frac{1}{2} +ix)$ when $\vert x \vert \to \infty$

I am currently looking for finding behaviour of the function $\vert \Gamma(\frac{1}{2}+ix) \vert$ when $x$ tends to $\infty$. I think I need to use the Stirling's approximation but I don't see how. ...
5
votes
3answers
190 views

Proving $\int\limits_{0}^{\infty}\frac{1-\text{e}^{-x}\cos(ax)}{x^{r+1}}\operatorname d\!x = \frac{\Gamma(1-r)}{r}(1-a^2)^{r/2} \cos(r \arctan(a))$

does anyone have an idea or a guess how to prove the following equation: $$\int\limits_{0}^{\infty}\frac{1-\text{e}^{-x}\cos(ax)}{x^{r+1}}\operatorname d\!x = ...
0
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0answers
58 views

How do i prove that the Gamma function is differentiable on its domain?

I have proved all of it's equivalence form (in wikipedia) of the Gamma function. That is: $\int_0^\infty t^{z-1} e^{-t} dt = \lim_{n\to\infty} \frac {n^z n!}{z(z+1)\cdots (z+n)} = ...