1
vote
1answer
58 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
10
votes
3answers
159 views

Scary contour integral, but is also an integral representation for $\Gamma$-function

This problem is supposed to be from an old Acta Mathematica volume I circa 1880's, and is attributed to Bourguet. By using a parabola with its focus on the origin as a contour, show that: ...
2
votes
1answer
58 views

Evaluation of the definite integral: $\int_0^{\pi/2}x\tan(x)^pdx$

The integral: $$J=\int_{0}^{\pi/2}x\,\tan^{p}\left(x\right)\,{\rm d}x$$ has the solution: ...
2
votes
1answer
62 views

Integral involving Gamma Function

I am solving the following integral: $$ \int_{-1}^{K} u^B e^{-u} du $$ The solution of the integral is a lower incomplete Gamma Function if -1 is replaced with 0. Can anybody help me in solving the ...
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
1answer
41 views

Evaluating improper integral expression

Can anybody please guide me in evaluating this expression, for my research work. I have tried a lot but in vain. Both the integrals involve gamma functions and even wolfram said that the time limit ...
0
votes
0answers
52 views

Solving integral that contain upper incomplete gamma function, exponential, and powers

I have this integration formula; $ f=\int\limits_{0}^{\infty}\frac{e^{-b~z}}{\sqrt{z}} \Big(\frac{\beta}{\beta+z}\Big)\Big(\frac{\beta+z}{z}\Big)^L ...
1
vote
2answers
116 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 ...
0
votes
1answer
46 views

Definite integral

So I was playing around with Euler's Reflection Formula ($\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin(\pi s)}$), trying to prove it with calculus, and was able to reduce $$ ...
1
vote
1answer
34 views

E[Log(x+a)] when x has gamma distribution

Is there a formula for this using built-in functions in matlab or mathematica like the Gamma functions or Ei's? $$\int_0^\infty \log(x+a)e^{-\alpha x}x^\beta dx. $$ Thanks.
7
votes
4answers
167 views

Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$ I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0. $$ I am ...
5
votes
2answers
167 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 = ...
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?
3
votes
0answers
77 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 ...
0
votes
1answer
203 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
46 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 ...
5
votes
3answers
205 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 = ...
3
votes
2answers
262 views

Fourier transform of complex Gamma function

I am wondering if it is known how to evaluate the Fourier transform of the complex Gamma function, i.e. $$ ...
4
votes
2answers
268 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= ...
7
votes
1answer
203 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
vote
1answer
70 views

Integral formula for Hypergeometric function

Going through some notes I seem to have used the following expression: $$\tag{1} \int_0^{\infty}u^k(1+u)^{n-k}(1+2u)^{-n-3}du=B(k+1, 2)\ _2F_1(n+3, k+1;k+3;-1)$$ where $k, n\in\{2,4,6,\ldots\}$, ...
4
votes
6answers
205 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? $$\int_{0}^{a}x^{n-1}e^{-x}dx$$
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 $$ ...
5
votes
1answer
149 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
4answers
141 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 $n\in \mathbb{N}$.
19
votes
2answers
356 views

Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$

I need to find a closed form for these nested definite integrals: $$I=\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$$ The inner integral can be ...
2
votes
1answer
153 views

Integral identity with square of Jacobi polynomial

This has stumped me for a while: I have a function $\zeta_k^S(x)$ that can be expressed using Jacobi polynomials $P_k^{(\alpha,\beta)}(x)$: ...
1
vote
1answer
53 views

Show that $\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$

The question asks to prove the identity: $$\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$$ where $n\in\mathbb{Z}$ I have no idea ...
4
votes
2answers
81 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: ...
3
votes
0answers
146 views

Is there a closed form expression for this integral?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
3
votes
1answer
149 views

How to relate this integral for the $\Gamma$ function to the defining integral of the $\Gamma$

In another question of mine at mse I had the detail of the two assumed identities $$ C_{b,p} = \int_0^\infty b^{x^p} dx $$ and $$ C_{b,p}={n! \over (- \beta)^n} $$ for some $ \small n \in \mathbb N $ ...
3
votes
1answer
1k 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 : $$ x=e^{u} $$ ...
0
votes
1answer
251 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
5answers
995 views

Calculate integrals involving gamma function

What are the usual ways to follow in order to solve the integrals given below? $$\begin{align*} I&=\int_0^1 \ln\Gamma(x)\,dx\\ J&=\int_0^1 x\ln\Gamma(x)\,dx \end{align*}$$
4
votes
2answers
228 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 ...
2
votes
1answer
248 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 ...
4
votes
2answers
739 views

Definition of the gamma function

I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ ...
2
votes
2answers
230 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 ...