2
votes
1answer
16 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 ...
0
votes
0answers
27 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: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
0
votes
1answer
35 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 &= ...
1
vote
1answer
49 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 ...
2
votes
2answers
45 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 $$\cos t=1-2\sqrt {u}$$
1
vote
1answer
23 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 ...
7
votes
2answers
148 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
votes
2answers
96 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 ...
0
votes
1answer
101 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.
1
vote
0answers
35 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
votes
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 ...
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 = ...
2
votes
1answer
67 views

How do I integrate $x^{\frac{3}{2}}e^{-x}$ from 0 to inf?

I have to evaluate the following expression : $$\int^{\infty}_{0} x^{\frac{3}{2}}e^{-x}$$ Wolfram|Alpha evaluates to $\frac{3\sqrt{\pi}}{4}$. I don't see how we got there. A hint would be helpful. ...
0
votes
2answers
64 views

resolving integral using gamma function

My book solves this integral \begin{equation} \int_0^{\infty} y^3e^{-ay} dy \end{equation} using gamma function as \begin{equation} \frac{1}{a^4}\Gamma(4) \end{equation} why is this true?
3
votes
1answer
135 views

An integral representation of the gamma function

An integral representation of the gamma function that would appear to be valid for all complex values of $z$ excluding the integers is $$ \Gamma(z) = \frac{2e}{\sin \pi z}\int_{0}^{\infty} ...
4
votes
2answers
189 views

How to solve this integral: $\int_{-1}^{1} x^k (1-x^2)^{(n/2)-2} \, dx$

How to solve this integral step by step: $$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx=??? $$ In my text book, it shows the result like below: $$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx= ...
0
votes
1answer
59 views

How to solve this gamma integral

Let we have the (p.d.f) of x which is: $$f(x)=\frac {\Gamma{(n-1)/2)}}{\Gamma{(1/2)} \Gamma{(n-2)/2)}}x(1-x^2)^{(n/2)-2}$$ then to find the $E(x) = $$\int_{-\infty}^{\infty} x *(f(x) dx$ ; ...
7
votes
1answer
176 views

Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
-1
votes
1answer
120 views

How to solve gamma function integral for 4!

I have been trying to test my knowledge of the gamma function by calculating 1! and 4!. I got the right result for 1! but I cannot get 4! analytically. To be more specific, I think I am not ...
2
votes
1answer
69 views

Some more parameter integrals

It seems that the following formulas hold : $$\int_{0}^{\infty} \frac{1}{\sqrt{x^{2n}+1}} dx = \frac{\Gamma(\frac{n-1}{2n})\Gamma(\frac{2n+1}{2n})}{\sqrt\pi}$$ for any integer $n > 1$ and ...
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
6answers
200 views

How to integrate $\int_{0}^{a}x^{n-1}e^{-x}dx$

We know that $$\int_{0}^{\infty }x^{n-1}e^{-x}dx = \Gamma (n)$$ But how do we integrate this? $$\int_{0}^{a}x^{n-1}e^{-x}dx$$
3
votes
1answer
95 views

Interesting integral involving $\Gamma (z)$.

Find the value of $$\int_0^\infty t^{x-1}e^{-\lambda t \cos(\theta)} \cos(\lambda t \sin (\theta)) dt$$ where $\lambda >0$, $x>0$, and ${-1\over 2}\pi < \theta < {1\over 2}\pi$ in terms of ...
3
votes
1answer
82 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
1answer
99 views

Gamma function in the sight of Lebesgue and Riemann integration.

I am taking a somewhat hard measure theory course and I was asked to prove this: a) Let $\alpha > 0$ be a real number. Prove that $$\Gamma(\alpha):=\int_0^\infty e^{-x}x^{\alpha-1}dx$$ exists. ...
11
votes
0answers
411 views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $ is in terms of the Barnes G function. $$ \int_{0}^{z} \ln \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
6
votes
2answers
131 views

Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
1
vote
0answers
96 views

Limiting behavior of an integral involving incomplete Gamma function

I am wondering about the limiting behavior as $k\rightarrow\infty$ of the following integral: $$I(k)=\frac{2^{-k/2}}{\Gamma(k/2)}\int_{f(k)}^\infty ...
1
vote
1answer
31 views

Relation between two gamma functions

Does anyone know the relation between these two gamma functions? 1st) Gamma[1 + c, a (1 + b)] 2nd) Gamma[c, a (1 + b)] The question is: may I write the 1st like the 2nd times something? thank you ...
1
vote
0answers
33 views

Show that $\sum\limits_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$. [duplicate]

Prove that for $x,y$ positive integers, $$\sum_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$$ One way is to use the beta-gamma functions relation: ...
1
vote
3answers
162 views

I'm looking for several ways to prove that $\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$

I'm looking for several ways to prove that $$\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$$ for $-2< Re(m)< 0$
5
votes
2answers
83 views

Using $\Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! n^z}{z(z+1)(z+2)\ldots(z+n)} \right]$ to prove the Weiestrass product.

I was searching the web quite thoroughly in the last two days. I was in paralytically looking for a rigorous proof using $$ \Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! ...
12
votes
4answers
389 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 $$\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx ...
4
votes
1answer
127 views

Integrate the beta function $B(x,y)$ over the unit square.

I was looking a bit around on the site and disovered (to my surprise) that $$ \int_0^1 \log \Gamma(x+t)\,\mathrm{d}x = \log \Bigl(\sqrt{2\pi}\Bigr) - t + t \log t $$ and using this and the fact ...
13
votes
7answers
614 views

Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. [duplicate]

Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how?
5
votes
1answer
139 views

Integration gamma and beta: $\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$

How can we evaluate the following integral? $$\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$$ I can't find anything to substitute because all of the trigonometric identities are in square form...
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 $$\int_0^{\infty}\frac{s^{-\alpha}}{1 + s}ds \le \Gamma(1 - \alpha)\Gamma(\alpha).$$ ...
1
vote
2answers
853 views

using gamma function to simplify integration

I have to evaluate $\int_0^1 x^2 \ln(\frac1x)^3 $.I used gamma function and used substitution $t=\ln (\frac {1}{x})^3$. In this I get to integrate from $1$ to $-\infty$ with a minus sign ...
3
votes
4answers
137 views

How to compute $\int^{\infty}_{0} t^{(\frac1n-1)}\cos t \,\mathrm{d}t$?

How to calculate the below integral? $$ \int^{\infty}_{0} \frac{\cos t}{t^{1-\frac{1}{n}}} \textrm{d}t = \frac{\pi}{2\sin(\frac{\pi}{2n})\Gamma(1-\frac{1}{n})} $$ where $n\in \mathbb{N}$.
4
votes
2answers
77 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
182 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
201 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
122 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
148 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
87 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.
13
votes
1answer
358 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 ...
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 ...
3
votes
2answers
448 views

Help evaluating a gamma function

I need to do a calculus review; I never felt fully confident with it and it keeps cropping up as I delve into statistics. Currently, I'm working through a some proof theory and basic analysis as a ...
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
217 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 ...