5
votes
1answer
88 views

Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function

How to prove this identity for $n>1/2$? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$
8
votes
2answers
83 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): ...
2
votes
1answer
56 views

Asymptotics of coefficients $[x^n] \frac{1}{\Gamma(1+x)}$ as $n$ is great

I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $. We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty ...
7
votes
4answers
154 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
0
votes
1answer
58 views

Evaluating $\dfrac{1}{\Gamma (r)}\int_{0}^{x}(x-t)^{\alpha -1}t^{\lambda}dt$ [closed]

How can I evaluate the following integral $$\frac1{\Gamma(r)}\int_0^x(x-t)^{\alpha-1}t^\lambda\ dt$$
1
vote
2answers
75 views

Compute $\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$ as a function of $\Gamma(n)$

Is it possible to compute this integral $$\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$$ as a function of complete gamma $\Gamma(n)$. If possible, I'm looking for a closed form solution. Thanks!
1
vote
2answers
121 views

Evaluating integral using Beta and Gamma functions.

$$\int_0^{\infty} x^{-3/2} (1 - e^{-x})\, dx$$ Evaluate the above integral with the help of Beta and Gamma functions. I'm badly stuck. I'm getting Gamma of a negative number and have no clue how ...
1
vote
2answers
133 views

Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
1
vote
2answers
52 views

Product of Gamma functions I

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{8} \Gamma\left( \frac{k}{8} \right) \end{align} and can it be shown that the product \begin{align} \prod_{k=1}^{16} ...
5
votes
2answers
169 views

Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$

$$ I:=\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}. $$ Thank you. The Gamma function is given by $\Gamma(n)=(n-1)!$ and its integral representation is $$ \Gamma(x)=\int_0^\infty t^{x-1} ...
1
vote
2answers
68 views

Evaluating $\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$

$\frac{d}{dx}\int_3^{x^2}e^{t^3}dt$ I suppose I don't fully understand the notation used within this problem. Using the second fundamental theorem of calculus: $\int_a^b f(x)dx = F(x)\bigr|_a^b = ...
0
votes
0answers
33 views

Summing gamma functions

Suppose I'm practicing penalties. Initially my goalscoring probability is $p$. As I miss a chance, the probability increases by constant value $q$. So the probability that I score my first goal at ...
1
vote
1answer
103 views

How to evaluate $\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$

How to evaluate $$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$$, where n is integer > 0? I know the gamma function formula will give $$ ...
2
votes
1answer
33 views

Getting from a product of gamma functions to a fraction answer

I am working on an assignment question for my Advanced Calculus course and am having great difficulty working it out. In order to try and understand this type of question/working, I have found a ...
17
votes
1answer
249 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
47 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct a probabilistic model, I have tried different methods of approach, by parts, substitution, but to no avail. Any help with this would be greatly appreciated!
2
votes
2answers
141 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 ...
9
votes
1answer
214 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
81 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
40 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
70 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
18 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
71 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
72 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
536 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
86 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 ...
3
votes
2answers
212 views

How prove this integral $ \int\limits_0^1{\int\limits_0^1{\ln\Gamma\left({x+{y^3}}\right)}}dxdy =-\frac{7}{{16}}+\frac{1}{2}\ln 2\pi$

show that $$ I=\int\limits_0^1{\int\limits_0^1{\ln\Gamma\left({x+{y^3}}\right)}}dxdy =-\frac{7}{{16}}+\frac{1}{2}\ln 2\pi$$ where $$\Gamma{(a)}=\int_{0}^{\infty}x^{a-1}e^{-x}dx$$ then ...
14
votes
2answers
186 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
140 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
311 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
78 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} ...
16
votes
6answers
556 views

Show that $\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16}$

I could prove it using the residues but I'm interested to have it in a different way (for example using Gamma/Beta or any other functions) to show that $$ ...
3
votes
2answers
82 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 $$\int_0^{\infty}\frac{s^{-\alpha}}{1 + s}ds \le \Gamma(1 - \alpha)\Gamma(\alpha).$$ ...
36
votes
1answer
1k 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
295 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
224 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
667 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
152 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
193 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
86 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
82 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
128 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
124 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
63 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
317 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
300 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
223 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
119 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
28 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
302 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 ...