16
votes
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?
0
votes
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 &= ...
2
votes
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 ...
8
votes
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} ...
0
votes
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 ...
1
vote
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. " ...
1
vote
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
vote
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) = ...
1
vote
1answer
60 views

How to derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n+ 1)$?

How can you derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n + 1)$? I have tried checking Wolfram Alpha for a step-by-step solution, but none is given. Moreover, of what is the second function, ...
1
vote
1answer
71 views

Let $a_n = \int\limits_{0}^{n} e^{-x^4} dx$. Does $\{ a_n \}_{n \rightarrow \infty}$ converge?

Let $a_n = \int\limits_{0}^{n} e^{-x^4} dx$. Does $\{ a_n \}_{n \rightarrow \infty}$ converge? $\{ a_n \} =\{ \int\limits_{0}^{1} e^{-x^4} dx, \int\limits_{0}^{2} e^{-x^4} dx, ..., ...
4
votes
2answers
412 views

Proof that the gamma function is an extension of the factorial function

I've already proved that $$\Gamma (n)= (n-1)!$$ but I don´t really know what else to do to verify that $\Gamma$ is an extension of the factorial function for real numbers (positive) Thank you! And ...
3
votes
1answer
80 views

Expressing integral as gamma function

I was trying to compute the expected value $E[X^k]$ for a Weibull distribution, and I encountered this integral $$\int_{-\infty}^\infty t^{b+k-1}e^{-ct^b}dt$$ where $k>0$ is an integer and ...
14
votes
2answers
179 views

Sum of $\Gamma(n+a) / \Gamma(n+b)$

If $a$ and $b$ are positive real numbers, such that $b > a + 1$, can we find the sum $$\sum_{n=0}^{\infty} \frac{\Gamma(n+a)}{\Gamma(n+b)}?$$ For example I have found that $$\sum_{n=0}^{\infty} ...
3
votes
2answers
137 views

How find $\Gamma{\left(\frac{8}{9}\right)}=\frac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$

show that $$\Gamma{\left(\dfrac{8}{9}\right)}=\dfrac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$$ where Gamma function:http://en.wikipedia.org/wiki/Gamma_function I found this problem is ...
11
votes
3answers
269 views

Gamma $\Gamma$ meets $\gamma$

I am looking for a proofs of the following limits: $$ \lim_{x \to \infty} \Gamma \left(1+\frac{1}{x} \right)^x = e^{-\gamma}. $$ I find this limit interesting as it relates the gamma function ...
0
votes
2answers
72 views

Why are the limits of integration of $\Gamma (z)$ the way they are?

$$\Gamma (z)=\int_0^\infty t^{z-1}e^{-t} dt$$ To make things easier, let's call this $G(z)$. If we take the derivative with respect to $t$ of both sides, we get $\frac {dG(z)}{dt}=\frac {d}{dt} ...
3
votes
2answers
80 views

Inequality with Gamma function: how to prove it?

Let $\alpha \in (0,1)$ and $\Gamma(\alpha) = \int_0^{\infty}s^{\alpha - 1}e^{-s}ds$. I would like to prove that $$\int_0^{\infty}\frac{s^{-\alpha}}{1 + s}ds \le \Gamma(1 - \alpha)\Gamma(\alpha).$$ ...
31
votes
1answer
994 views

Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?

Is it possible to simplify this expression? $$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$ Is there a systematic ...
3
votes
2answers
258 views

Evaluate integral in terms of Gamma function

Need help evaluating the following integral in terms of the gamma function where gamma function is: $$\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}dt.$$ The integral is the following: $$\large{\int_0^\infty ...
15
votes
1answer
200 views

Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$

Working on this conjecture, I found its corollary, which is also supported by numeric caclulations up to at least $10^5$ decimal digits: ...
33
votes
5answers
650 views

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$

Is there a closed form for the following infinite product? $$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
1
vote
1answer
143 views

Gamma function question

Gamma function is also known as generalized factorial function . why does the term "generalized" have been used ? Again, why is the Gamma function called Euler's second integral?
4
votes
2answers
181 views

About the Beta function : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$.

Find the value of : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$, where $\text{B}(x,y)$ is the Beta function. Why do I need this ? Because I want to calculate : $$ \int\limits_{ - \infty }^\infty ...
2
votes
1answer
77 views

The rate of increase of the Gamma Function over real numbers

If $$ x_1 > x_2 > 0$$ and $$\Delta{x}>0$$ does it follow that: $$\ln\Gamma(x_1 + \Delta{x}) - \ln\Gamma(x_1) \ge \ln\Gamma(x_2 + \Delta{x}) - \ln\Gamma(x_2)$$ Would it be enough to show ...
3
votes
1answer
76 views

Solve $-B \ln y -A y \ln y + A y- A =0$ for $y$

I would like to know if there is a (preferably closed-form) solution for $-B \ln y -A y \ln y + A y- A =0$ for $y$ Where $A, B \in \mathbb{R}^{+}$. I have reasons to think there isn't a closed form ...
2
votes
2answers
123 views

Solve $B \ln y +A y \ln y + A y-A =0$ for $y$

I would like to know if there is a (preferably closed-form) solution for $B \ln y +A y \ln y + A y- A =0$ for $y$ Where $A, B \in \mathbb{R}^{+}$. I have reasons to think there isn't a closed form ...
5
votes
3answers
121 views

Integral in $\mathbb R^3$ and $\Gamma$-function

How one can show this equality? $$ \iiint_V x^{a-1}y^{b-1}z^{c-1}\,dxdydz = \dfrac{\Gamma(a)\Gamma(b)\Gamma(c)}{\Gamma(a+b+c+1)}, $$ where $V$ is simplex $x\geqslant0, y\geqslant0, z\geqslant0, ...
3
votes
2answers
61 views

Minimize the coefficient of asymptotic normality

I need to minimize this complicated function: $$\dfrac{\Gamma(2x+1)-\Gamma(x+1)^2}{x^2 \Gamma(x+1)^2}$$ (For people wondering where this function came from — it's the (multiple of) coefficient of ...
10
votes
3answers
302 views

Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$

Does anybody know how to prove this identity? $$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma ...
6
votes
1answer
284 views

Prove that $\int_{0}^{\infty }\frac{x^{a-3/2}dx}{[ x^2+( b^2-2)x+1]^a}=b^{1-2a}\frac{\Gamma(1/2)\Gamma(a-1/2)}{\Gamma(a)}$

How can one prove that $$I\left( a,b \right)= \int_{0}^{\infty }\frac{x^{a-\frac{3}{2}}dx}{\left[ x^2+\left( b^2-2 \right)x+1 \right]^a}=b^{1-2a}\frac{\Gamma \left( \frac{1}{2} \right)\Gamma \left( ...
10
votes
2answers
215 views

A $\log$ integral with a parameter

Prove that: $$\int_0^\infty \frac{\ln x}{x^a+1}\;\text{d}x=-\left( \frac{\pi }{a} \right)\cot \left( \frac{\pi }{a} \right)\csc \left( \frac{\pi }{a} \right),\ \ a>1$$ For this one I consider to ...
2
votes
2answers
117 views

About the use of Stirling approximation

How to prove this inequality: $$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$ Sry I forgot to mention that $x>300$
1
vote
0answers
27 views

Quick question on the simplification of digamma series

How to simplify : $$\sum\limits_{k=1}^{\infty }{\frac{{{\left( -1 \right)}^{k-1}}}{k}\left( \psi \left( \frac{k}{2\left( 2+\sqrt{3} \right)}+1 \right)-\psi \left( \frac{k}{2\left( 2+\sqrt{3} ...
1
vote
0answers
296 views

$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$, where $s$, $r$ and $m$>1 are positive integers. My question is whether a closed form ...
0
votes
1answer
240 views

Solve in terms of the Gamma function

Show: \begin{align*} \int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\,\mathrm d x &=\frac{\sqrt \pi}{4}\left(\frac{\Gamma ...
8
votes
1answer
789 views

Integral $\int_0^1\sqrt{1-x^4}dx$

I am asked to show $\int_0^1\sqrt{1-x^4}dx=\frac{\{\Gamma(1/4)\}^2}{6\sqrt{2\pi}}$. I know the gamma function is defined by $\Gamma(n)=\int_0^\infty x^{n-1}e^{-x}dx$. I tried to substituted ...
0
votes
2answers
498 views

what are the properties of gamma function? [closed]

In mathematics, the gamma function (represented by the capital Greek letter $\Gamma$) is an extension of the factorial function, example: $\Gamma(x)$, $\Gamma(ix)$. What are the physical properties ...
8
votes
5answers
904 views

Calculate integrals involving gamma function

What are the usual ways to follow in order to solve the integrals given below? $$\begin{align*} I&=\int_0^1 \ln\Gamma(x)\,dx\\ J&=\int_0^1 x\ln\Gamma(x)\,dx \end{align*}$$
8
votes
1answer
511 views

Question involving the Gamma function

I'm trying to prove that for $p,q>0$, we have $$\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$$ The hint given suggest that we express $\Gamma(p)\Gamma(q)$ as a double ...
4
votes
2answers
218 views

Computing $ \int_0^{\infty} \frac{ \sqrt [3] {x+1}-\sqrt [3] {x}}{\sqrt{x}} \mathrm dx$

I would like to show that $$ \int_0^{\infty} \frac{ \sqrt [3] {x+1}-\sqrt [3] {x}}{\sqrt{x}} \mathrm dx = \frac{2\sqrt{\pi} \Gamma(\frac{1}{6})}{5 \Gamma(\frac{2}{3})}$$ thanks to the beta function ...
4
votes
1answer
133 views

Convergence of this integral [duplicate]

Possible Duplicate: Some questions about the gamma function My statistics text book prescribed by my school states that the integral $$\Gamma(n)=\int_{0}^{\infty}e^{-x}x^{n-1}dx$$ is ...
2
votes
1answer
238 views

a gamma integral

any idea how to do this integral ? $$\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt$$ $j$ is a positive integer. $k$ is a constant - not necessarily ...
5
votes
3answers
320 views

Proof that $Γ'(1) = -γ$?

I know that $Γ'(1) = -γ$, but how does one prove this? Starting from the basics, we have that: $$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$ How do we differentiate this? How do we then find that ...
6
votes
4answers
247 views

Gamma identity $\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p})dx=\Gamma(p+1)$

I ran across what appears to be another Gamma identity. Show that $$\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p}) \,\mathrm dx=\Gamma(p+1)=p!$$ I tried several different subs and ...
5
votes
1answer
162 views

How to Express $\Gamma (x)$ in terms of $\cos$

I'm trying to figure out how to express the integral $$\int\limits_{0}^{\infty} \cos(x) \times x^{-a} \rm{dx}$$ as $$\cos\frac{\pi-a\pi}{2}\times \int\limits_{0}^{\infty} e^{-x} x^{-a} \ \rm{dx} ...
2
votes
2answers
226 views

modified gamma integral

I have the following integral $$\int_0^{+\infty} t^{z-1} e^{-t} \frac1{(kt + 1)^s}\mathrm dt$$ where $k>0, s > 0$. How would you suggest to solve it? Without $\frac1{(kt + 1)^s}$ it would be ...
7
votes
1answer
277 views

Is there a gamma-like function for the q-factorial?

I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in ...
7
votes
2answers
701 views

Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?

The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some properties of the gamma function: $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$ $\Gamma (1)=\Gamma ...