Tagged Questions
3
votes
2answers
52 views
Definite integral formula from wikipedia
I need to solve a certain definite integral, and several places (for example Wikipedia) I've come across the following formula:
...
4
votes
2answers
96 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 : $$
\int\limits_{ - \infty }^\infty ...
0
votes
0answers
72 views
Tight Upper/Lower bound for Incomplete Gamma function
Does anyone know of any tight upper/lower bound for incomplete Gamma functions? i.e either of the following functions:
$$
\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t
$$
or
$$
\gamma(s,x) ...
5
votes
3answers
91 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, ...
1
vote
2answers
64 views
$\Gamma ( \alpha)$ function
Gamma function of $\alpha$ is defined as
$\Gamma \left( \alpha \right) = \int\limits_0^\infty {y^{\alpha - 1} e^{ - y} dy}$
Gamma function exist for $ \alpha > 0 $ why???
I think the reason is ...
4
votes
1answer
71 views
Strategy for Improper Integrals Related to the Beta Function 2
I am looking for the solution of the following integral
$$\int_0^1 y^k \log\left(1+\left(\frac y{1-y}\right)^a\right)dy,\quad a>0 $$
I really appreciate it if any one can help.
12
votes
1answer
187 views
Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$
I am asking this question out of curiosity.
$$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha ...
8
votes
2answers
155 views
Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$
Does anybody know how to prove this identity?
$$\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 ...
1
vote
0answers
165 views
Prove this integral (about gamma function)
Prove that :
$$I\left( a\ ,\ b \right)=\int_{0}^{\infty }{\frac{{{x}^{a-\frac{3}{2}}}}{{{\left( {{x}^{2}}+\left( {{b}^{2}}-2 \right)x+1 \right)}^{a}}}\text{d}x}={{b}^{1-2a}}\frac{\Gamma \left( ...
10
votes
2answers
191 views
A $\log$ integral with a parameter
Prove that:
$$\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$$
For this one I consider to ...
1
vote
0answers
23 views
Quick question on the simplification of digamma series
How to simplify :
$$\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} ...
2
votes
1answer
217 views
How to calculate these gamma functions?
Equation : $$\int _{0}^{\infty }x^{n}e^{-x}dx=n!=\Gamma(n+1) $$
1) $$ \int _{0}^{1}x^{2}\left( \ln \dfrac {1} {x}\right) ^{3}dx $$
2) $$\int _{0}^{1}\sqrt[3] {\ln x}dx $$
Hint : $$ x=e^{u} $$
...
3
votes
4answers
148 views
Integrating $\int_0^{\infty} u^n e^{-u} du $
I have to work out the integral of
$$
I(n):=\int_0^{\infty} u^n e^{-u} du
$$
Somehow, the answer goes to
$$
I(n) = nI(n - 1)$$
and then using the Gamma function, this gives $I(n) = n!$
What I do ...
1
vote
1answer
37 views
Integral of Singh-Maddala cdf
What is the integral or maybe a suggested technique to find
$$\int_0^x \left(1+\left(\frac{t}{b}\right)^a\right)^{-q} d t,$$
$a,b,q\in \mathbb{R}:a\cdot q>1,b>0.$
?
1
vote
0answers
260 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
203 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 ...
0
votes
2answers
394 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 ...
6
votes
4answers
624 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
311 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 ...
2
votes
1answer
182 views
Strategy for Improper Integrals Related to the Beta Function
I'm interested in whether or not integrals of the form $I (\alpha , \beta , k) = \displaystyle\int_0^1 \log^k \left(\frac{x}{1-x}\right) x^{\alpha -1} (1-x)^{\beta -1 }dx,$ with $k\in \mathbb{N}, ...
0
votes
0answers
124 views
Can one find the inverse function for a combination of imcomplete gamma functions?
The original function was defined as $f(z)=y=\left( \Gamma \left( k,{\frac {z-b}{\theta}} \right) -\Gamma \left( k,{\frac {z-a}{\theta}} \right) \right) \left( \Gamma \left( k \right) \right) ...
1
vote
2answers
145 views
Resummation of the Series
I wonder if there is a way to get resummation of this series? By this way , i am trying to get the integral representation of this series, it could be by Gamma function.
$$\sum _{k=0}^{\infty} \left ...
5
votes
3answers
261 views
Proof that $Γ'(1) = -γ$?
I know that $Γ'(1) = -γ$, but how does one prove this?
Starting from the basics, we have that:
$$Γ(x) = \int_0^\infty e^{-t} t^{x-1} dt$$
How do we differentiate this? How do we then find that
...
6
votes
4answers
207 views
Gamma identity $\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p})dx=\Gamma(p+1)$
I ran across what appears to be another Gamma identity.
Show that $$\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p}) \,\mathrm dx=\Gamma(p+1)=p!$$
I tried several different subs and ...
5
votes
2answers
566 views
Convergence of $\Gamma(p)$ for $0<p\leq 1$ and divergence for $p \leq0$.
Can someone show me a proof or any clear resource about convergence of gamma function for values of $p$ less than zero.
If possible I need proofs using integration by parts.
My problem evaluating ...
1
vote
2answers
114 views
How to integrate this equation
all.
I am trying integrate this equation(gamma density).
$$\int\limits_0^\infty \frac{1}{\sqrt{\left| x-1 \right|}}\exp \left( -\left| 1-x \right| \right) \;dx$$
What I have done is split it into 2 ...
2
votes
1answer
180 views
Intuitive limit. Integration and $\gamma(n,x)$
It isn't hard to prove that:
$$\int_0^x e^{-t} {t^n} dt = n! \cdot e^{-x}\left( e^x-\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$
Or put in a different way:
$$\int_0^x e^{-t} \frac{t^n}{n!} dt = ...
2
votes
1answer
199 views
About integration related to the gamma function
I would like to compute the integral
$$
\int_{0}^{\infty}\frac{1}{\sqrt{2t}}e^{-\frac{1}{2t}}dt
$$
which wolfram alpha says that it does not converge. However by letting $x=1/2t$ I get ...
4
votes
2answers
212 views
Integral form of $\Gamma (x)$
While trying to represent the poles and analytic continuation of $\Gamma (x)$ , the author uses the following equality:
$$\int_{0}^{1}t^{x-1}e^{-t}dt=\sum_{n=0}^{+\infty}\frac{{(-1)}^{n}}{(n+x)n!}.$$
...
7
votes
2answers
324 views
Integration of powers of the $\sin x$
I have to evalute
$$\int_0^{\frac{\pi}{2}}(\sin x)^z\ dx.$$
I put this integral in Wolfram Alpha, and the result is
...
0
votes
1answer
138 views
Coordinate scaling in incomplete gamma function integral
I'm faced with the integral
$$\mathcal{I} = \int_0^\infty \mathrm d x \; e^{-\beta \, e^x - \mu x} \;,\quad \Re(\beta) > 0 \;.$$
The solution can be looked up. It reads
$$\mathcal{I} = \beta^\mu ...
1
vote
1answer
179 views
Integral of product of Gamma densities over probability simplex
I want to find a simpler form or closed form for the following integral:
$$
\int_A \,\prod_{t=1}^T f_{\Gamma(a,\theta_t)}(x_t) \,d\mathbf{x}
$$
where $A$ is the simplex
$$
A = \{\mathbf{x} \in ...
0
votes
0answers
129 views
Determining well-definedness for functions
How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function?
Firstly:
$$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$
...
2
votes
2answers
198 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
2answers
481 views
Limit of Gamma integrals
The following seems to hold in numerical simulations, is it true?
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$
It's a combination of two known integrals
...
6
votes
2answers
609 views
An elementary way of simplifying a trigonometric triple integral?
By stressing my manipulative powers and a bit of help from Mathematica I was able to show that the triple integral
...
