1
vote
0answers
42 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
0
votes
0answers
66 views

Integrate $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school IMO training ...
3
votes
0answers
68 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
0answers
30 views

Help with a double integral involving the modified Bessel function of the first kind

Consider the function $f(t)$ given by the double integral: $$f(t)=\frac{2}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}\frac{I_{0}\left(2\sqrt{y x} \right )-1}{e^{y}-1}\cos(xt)dydx$$ Where $I_{0}(\cdot)$ is ...
6
votes
1answer
66 views

Integral $I=\int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx$

Hello can you please help me solve this integral $$ \int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx=-\frac{2\pi}{3}(\pi^2+24\ln 2). $$ I am trying to work through all logarithmic integrals. Note, ...
5
votes
1answer
89 views

Integral computation of $\int_0^\pi \mathrm d t \sin(a\cos t/2) \mathrm{sinh}(b\sin t/2)$

I'm having trouble computing an integral. $$ I=\int_0^1 \frac{\mathrm{d}x}{2x(1-x)}\left(x-\cosh\left(\frac{t\sqrt{1-x}}{\tau}\right)+\sqrt{1-x}\text{ ...
1
vote
0answers
74 views

solution of another definite integral

Does the following integral converge or not? \begin{align} && \sum_{k=0}^{\infty} (-\varphi)^k \binom{\frac1\varphi+k}{k}\int_{-\infty}^\infty\beta x^n e^{-\beta x(k+1)}dx&& ...
1
vote
0answers
38 views

finding value of definite integral

Does the following definite integral exist here $\alpha > 0$ and $n$ is positive integer.
1
vote
3answers
49 views

search for closed form solution of definite integral

Integrate/hint for this definite integral $$\int_0^\infty(\log\theta)^n\frac{1}{\theta^{k+2}}\text{d}\theta,$$ where $n$ and $k$ are positive integers. It is a simplified form of my earlier question ...
6
votes
0answers
90 views

analytic solution to definte integral

I am looking for Analytic solution to a definite integral. Or an approriate transformation to apply. the conditions on alpha, beta being positive real numbers while n is positive integer.the integral ...
0
votes
1answer
39 views

Taylor series of a rational function

I am facing some complicated integral, which part of it is $$\frac{z^{M-1}}{(1+(\eta z)^n)^p}$$ I think if I find the taylor series of this part the integral might be solved. So, can someone help me ...
11
votes
1answer
145 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
0
votes
1answer
99 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.
0
votes
0answers
33 views

Help with taylor series as part of an integral involving gamma function

I am facing some strange problem regarding the Taylor series for this function: $$\frac{1}{(1+(\eta z)^n)^p} = ...
16
votes
1answer
466 views

Fourier Transform $J^3_0(x)$.

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $$ analytically. We can also write $$ \mathcal{I}(\omega)=\mathcal{FT}\big(J^3_0(x)\big) ...
0
votes
0answers
25 views

integral involving hypergeometric function

I've obtained that the eigenfunctions of a certain Sturm-Liouville problem are: $$ \phi(x,\lambda) = C\cdot(1/x)^{-1/2\pm i\lambda}\Psi(-1/2\pm i\lambda, 1\pm2i\lambda,1/x), $$ where $C$ is a ...
14
votes
2answers
147 views

Integral $\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz$ involving Dawson's integrals

I need you help with evaluating this integral: $$I=\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz,\tag1$$ where $F(x)$ represents Dawson's integral: $$F(x)=e^{-x^2}\int_0^x ...
4
votes
1answer
84 views

Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$

I'm interested in the following integral: $$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$ where $K(z)$ is the complete elliptic integral of the 1ˢᵗ ...
3
votes
1answer
58 views

Solution of definite integrals involving incomplete Gamma function

The solution of the integral $$\int_0^{\infty}e^{-\beta x}\gamma(\nu,\alpha \sqrt x)dx $$ is given as ...
4
votes
2answers
122 views

Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral

Calculate $$ I_1:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx, \ p \geq 1. $$ I am trying to solve this integral $I_1$. I know how to solve a related integral $I_2$ $$ I_2:=\int_0^1 \frac{\ln \ln ...
2
votes
0answers
117 views

integral $I=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
1
vote
0answers
39 views

Solution of some Bessel integrals

The solution of the integration $\int_0^\infty e^{-\alpha x}J_v(\beta x)x^{\mu-1}dx$ is given in a standard form. Can I use the same result when the upper limit of the integration is finite? The ...
28
votes
3answers
434 views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ but have problems ...
10
votes
1answer
146 views

How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
4
votes
2answers
81 views

Does anyone know how to calculate the following integral?

Consider the function (coming from a joint probability density): $$ f(x,y) = \frac{1}{y}e^{-y-\frac{x}{y}}. $$ I want to evaluate the definite integral (marginal): $$ F(x) = \int_0^\infty f(x,y)\,dy. ...
0
votes
0answers
33 views

help with complicated modified bessel function integral

I am trying to address the following complicated integral $$\int_0^{\infty} x^{m-1} e^{-(ax^2+bx+c)}I_v(kx)\text{d}x,$$ Where $I_v(x)$ is a modified Bessel function of the first kind. I did try to ...
11
votes
1answer
244 views

How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
0
votes
1answer
49 views

Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.

I encountered the above when dealing with the Bessel functions of the first kind, $J_n(z)$, specifically $n=0$. Using the differential-equation definition of the Bessel function, I obtained the above ...
1
vote
0answers
29 views

Integral with incomplete gamma function and Modified Bessel Function

Can somebody suggest a technique to integrate this? all parameters (m, beta, v, k, A, B and n) are positive real constants
0
votes
0answers
45 views

Integral Involving Trigonometric Functions and Exponential (Related to Marcum Q-function)

I want to solve this integral $$ \int_{0}^{\infty}\int_{0}^{2\pi}\exp(-ar^2)\exp(r\,b(\cos\theta+\sin\theta))r^{m}\cos^{m}(2\theta)d\theta \,dr,$$ where $a$ and $b$ are constants. I know how to ...
1
vote
0answers
34 views

A nice form for this Beta-like integral $\int_0^\frac{\pi}{2} \sin^\alpha(n t) \cos^\beta(t)dt$?

I was differentiating a completely different integral when I came across $$I(\alpha,\beta,n)=\int_0^\frac{\pi}{2} \sin^\alpha(nt) \cos^\beta(t)dt.$$ Evidently, $I(\alpha, \beta,1)= \frac{1}{2} B ...
0
votes
0answers
37 views

Interesting integral with Modified Bessel Function, Gamma Function

Is there anyway to integrate this monster? m, beta, v, k, A, B, and n are real positive constants.
0
votes
3answers
97 views

Gaussian-Like integral

What is the integral of this? $$\int_0^\infty xe^{-(ax^2+bx)}\,\mathrm{d}x$$ $a$ and $b$ are positive integers.
2
votes
1answer
40 views

Integral with the incomplete upper gamma function

Can anyone help me integrate this? $$\int_0^1 \frac{1}{x^{1/p}} \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{m/n-1} \Gamma\;\left(A, \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{1/n}\right) ...
1
vote
0answers
21 views

integral of incomplete gamma function and other functions

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
3
votes
2answers
76 views

Is This a Bessel Function?

Is the function $$y(x) = c \int_{-1}^1 \cos(xt)(1-t^2)^{n-\tfrac{1}{2}}dt = c \sum \tfrac{(-1)^m x^{2m}}{(2m)!} \int_{-1}^1 t^{2m}(1-t^2)^{n-\tfrac{1}{2}}dt$$ given here a bessel function? It ...
1
vote
0answers
42 views

Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} ...
5
votes
2answers
248 views

Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$

Is there a closed form expression for the definite integral $$I=\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$$ for $a<b<c<d$? Mathematica 9.0 can do it for special cases using ...
1
vote
1answer
52 views

integral with gaussian function

I am trying to evaluate the following integral: $$ \int_0^\infty{z^{m-1}\over\left[1+\left(\eta z\right)^n\right]^p}e^{-(z-b)^2\over c}\,{\rm d}z, $$ where the integration is w.r.t. to $z$, and the ...
2
votes
0answers
93 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
2
votes
0answers
34 views

comparison of two integrals

Let $n \in N$. How to compare two integrals: $$ I_1=\int_0^{\infty}\left(\frac{\sin t}{t}\right)^n dt \quad \text{and} \quad I_2=\int_0^{\pi}\left(\frac{\sin t}{t}\right)^n dt\,\, ? $$ I've beet ...
8
votes
1answer
112 views

Integral in $n-$dimensional euclidean space

I want to calculate this integral in $n$-dimensional euclidean space. $$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, ...
2
votes
0answers
20 views

Solving $\int_0^{+\infty}\,\frac{k}{k^2-\alpha^2}\,J_1(kR)\,J_1(ka)\,\exp(-d^2k^2)\,dk$

I approached this integral to be solved: $$\int_0^{+\infty}\,\frac{k}{k^2-\alpha^2}\,J_1(kR)\,J_1(ka)\,\exp(-d^2k^2)\,dk$$ where $J_1(x)$ is the Bessel function of the first kind and $R,a,d,\alpha$ ...
2
votes
1answer
28 views

Solving $\int_0^{+\infty}\,k^{-1}\,J_1(kR)\,J_1(ka)\,\exp(-d^2k^2)\,dk$

I am trying to solve the following integral: $$I(a,R,d)=\int_0^{+\infty}\,k^{-1}\,J_1(kR)\,J_1(ka)\,\exp(-d^2k^2)\,dk$$ where $J_1(x)$ is the Bessel function of the first kind and $R,a,d$ have real ...
5
votes
3answers
120 views

How to solve $\int_0^{+\infty}\,ax\,J_0(ax)\,dx$

From some equalities I ended up with understanding that: $$\int_0^{+\infty}\,ax\,J_0(ax)\,dx = 1$$ with $J_0(ax)$ the bessel function of the first kind and $a>0$. But I don't know how to ...
8
votes
3answers
215 views

Evaluating $\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$

I was trying to find a closed form for $$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz = -2.454199511\cdots$$ where $\text{Li}_2(z)$ is the ...
2
votes
1answer
112 views

help with exponential integral and square root

can somebody help me integrating this: where m, p and q are positive constants. I tired change of variables and searched for the solution but could not find it. Thanks Note: this is the result ...
2
votes
1answer
69 views

integral of modified bessel function of 2nd type

I need some help on a possible way to integrate this: $$ \int_0^\infty{x^{m-1}\mathrm{e}^{-\lambda x}\left[\frac{\operatorname{K}_\nu\left(b\sqrt{\alpha+\beta x}\right)}{\left(b\sqrt{\alpha+\beta ...
6
votes
1answer
178 views

Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?

In evaluating integrals like (link to another example) $$I=\int_0^1\frac{\log(x) \log^2(1-x)dx}{x}$$ one can make the substitution $x=\sin^2(\theta)$ to obtain ...
1
vote
1answer
186 views

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have ...