2
votes
0answers
42 views

An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
7
votes
1answer
76 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
3
votes
0answers
27 views

integral over product of two bessel functions at discontinuity

The Weber-Schafheitlin integral $$ \int_{0}^{\infty}\frac{J_{\mu}(a t)J_{\nu}(bt)}{ t^{\lambda}} $$ where $J_{\mu}(x)$'s are Bessel functions of the first kind, can have delta function singularities ...
14
votes
1answer
264 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
1
vote
1answer
102 views

Trying to solve $\int{-2\exp{\left(z\cos^2 \theta \frac{\left(a^2 - 1\right)}{2a^2}\right)}}d\theta$

I am trying to solve this integral which has come up as part of some other work, but it is proving to be much harder than I had originally thought. For $0 < |a| \le 1$ being some constant, I am ...
4
votes
3answers
68 views

How to show that $\int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt = \pi^{1/2} \frac{ \Gamma(s-1/2)}{\Gamma(s)}$

I want to compute the integral $$ \int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt $$ for $s \in \mathbb{C}$ such that the integral converges ($\mathrm{Re}(s) > 1/2$ I think) in terms of the Gamma ...
1
vote
1answer
51 views

Equailty involving Elliptic integrals and hypergeometric function

How to prove the following $$\,_2F_1\left(-1/2,-1/2,1,k^2 \right)=\frac{2}{\pi}\left(2E+(k^2-1)K \right)$$ where we define The complete integral of first kind $$K=K(k) = \int^1_0 ...
3
votes
0answers
34 views

integral involving hypergeometric function $\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{y})}{y}\,dy$

I arrived at the following result $$\tag{1}\int^\infty_0 z^{p-1} E^2(z)\,dz=\frac{\Gamma(p)}{p}\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{z})}{z}\,dz$$ where the exponential integral $E(z)$ is defined ...
4
votes
2answers
171 views

Definite integral $\int_{R_0}^{R}\frac{dr}{r^2\sqrt{\frac{R_0-R_S}{R_0^3}-\frac{1}{r^2}\left(1-\frac{R_{s}}{r}\right)}}$

In general relativity, null geodesics (in the unbounded case) can be written under the following form : ...
0
votes
1answer
171 views

Evaluation of definite integral using complex analysis

I want to evaluate the following indefinite integral $$ \int_0^{\infty} x^{p - 1} \cos (ax) dx$$ where $0 < p < 1$ and $a > 0$. I was considering the function $f(z) = z^{p - 1} e^{iaz}$ and ...
0
votes
0answers
41 views

Incomplete gamma function and hypergeometric function to Meijer-G

can somebody help me to convert the incomplete gamma function and the hypergeometric function (in the forms shown below and as a function of z) into a form of Meijer-G function?
1
vote
2answers
213 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
3answers
72 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
1answer
154 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 ...
0
votes
0answers
43 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} = ...
3
votes
1answer
125 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 ...
1
vote
0answers
48 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 ...
0
votes
3answers
113 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
144 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
2
votes
1answer
53 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
30 views

How to find $\int_0^\infty \frac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$

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 ...
1
vote
1answer
57 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
109 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
1answer
122 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 ...
3
votes
0answers
67 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
2
votes
1answer
94 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 ...
1
vote
1answer
75 views

How to evaluate the integral involving two Bessel functions and following elementary function?

Let's have the integral $$ \int \limits_{0}^{\infty} J_{0}(kr) J_{0}(kr')\frac{kdk}{k^2 + a^2}. $$ How to evaluate it? I failed when was trying using integral representations for the Bessel ...
3
votes
2answers
95 views

How to evaluate following integral

I tried to evaluate integral $$ I = \int \limits_{0}^{\infty}\frac{J_{\nu}(bx)x^{\nu + 1}dx}{(x^{2} + a^{2})^{\mu + 1}}, $$ but I failed. I can't represent Bessel function as a series, because ...
0
votes
1answer
73 views

Closed-form Solution of an Integral Equation

Recently, I get stuck in an equation, and I would like to obtain an closed-form solution of an equation, which is given as $$ \int_1^\infty\frac{x}{t^\alpha-x}dt=c, $$ where $\alpha>1$ and $c>0$ ...
0
votes
0answers
77 views

Limit of the integral, integral of the limit (no dominated convergence)

I'm working on the integral: $\lim_{x \rightarrow 0} \int_0^T \frac{1}{\sqrt{\pi t}} \exp \left(-\frac{x^2}{t}-t\right) \left(1-\frac{t}{T} \right) \mbox{d}t$ I'ld like to inverse the integral and ...
1
vote
0answers
118 views

Conditional expectations of the beta distribution in closed form?

I have a variable distributed Beta(a, b). I need to take expectations of an exponential function. $e^{c z}$. Forgetting constants of proportionality throughout, the expectation is: $\int_0^1 e^{c z} ...
0
votes
0answers
69 views

A bessel function type integral

According to http://people.math.sfu.ca/~cbm/aands/page_376.htm, I know that $\int_0^\infty e^{-z\cosh t}\cosh vt~dt=K_v(z)$ and $\int_0^\infty e^{-z\cosh ...
2
votes
1answer
108 views

Integral of function with a pretty long name

I am looking for a compact result of this integral: $$\int_R^\infty k_n(x) \, dx,$$ where $k_n$ is the modified spherical Bessel function of the second kind (explanation to this function). ...
2
votes
1answer
119 views

Oscillating integral

I want to calculate $$ \int _0^\infty e^{-iyx}\sqrt{x(x+2)}\, dx $$ in the sense of distributions, at least for $y\ne 0$. Now, I happen to know the following integral representation for the modified ...
1
vote
1answer
1k views

A method for calculating this integral hermite polynomials

I need proof this, $\int_{-\infty}^{\infty}e^{-x^2}H_n^{2}(x)x^2dx=2^nn!\sqrt{\pi}(n+\frac{1}{2})$ This is the idea: Multiply ...
13
votes
7answers
659 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?
3
votes
2answers
192 views

Is there a formula for this integral

Is there a formula for the following integral? $$I(a,b)=\int_0^1 t^{-3/2}(1-t)^{-1/2}\exp\left(-\frac{a^2}{t}-\frac{b^2}{1-t} \right)dt$$ where $a,b$ are non-zero real numbers.
13
votes
3answers
681 views

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$ We can prove using the Beta-Function identity that $$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma ...
2
votes
1answer
105 views

Numerical methods for integrals involving product of Bessel functions of the first kind (1st order)

I am looking for the best (in terms of low computation times) numerical methods for calculating the following integrals: $$\int_0^{\infty}\,f(k)\,J_1(ak)\,J_1(bk)\,dk$$ with for instance ...
4
votes
1answer
248 views

Integral involving bessel function/gaussian/rational function

I'd like to solve: $$\int_0^{\infty}\quad J_1(ak)\,\frac{b+k^2}{(k-\alpha_1)(k-\alpha_2)(k-\alpha_3)(k-\alpha_4)}\,\exp(-ck^2)\,\,dk$$ Is there any specific rule for it? Thanks!
3
votes
2answers
303 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 ...
19
votes
1answer
238 views

What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

Let $\operatorname{li} x$ denote the logarithmic integral: $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Is it possible to find a closed form of the following integral? $$\int_0^1\ln(-\ln x) ...
2
votes
0answers
133 views

Integral involving bessel functions, exponential and ratio of polynomials

I need to solve this integral: $$\int_0^{+\infty}\quad\frac{k}{k^2-\alpha^2}\,J_1(2\pi R k)J_1(ak)\,\exp(-4\pi^2\omega^2k^2)\, dk$$ where $\omega,\,R,\,a>0$ and $J_1(a\alpha)=0$. Thanks!
33
votes
0answers
734 views

Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
6
votes
2answers
253 views

How to prove a generalized integral identity

$$ \int_{0}^{\infty }\frac{t}{(e^{2\pi t}-1)(1+t^{2})}dt=-\frac{1}{4}+\frac{\gamma}{2} $$ where $\gamma$ = Euler Gamma $$ \int_{0}^{\infty }\frac{t}{( e^{2\pi t}-1)(1+t^{2}) ^{2}}dt=\frac{\pi^2}{24} ...
3
votes
1answer
140 views

Improper integral and special functions

I'd like to have an expression of the following integral: $$\int_0^{+\infty} \left(\sqrt{1+x^4} - x^2\right) dx$$ in terms of some special functions (but not in the form given by Wolfram Alpha).
18
votes
1answer
224 views

Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$

I need help with solving this integral: $$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$ where $\text{Li}_{s}(z)$ is the polylogarithm.
19
votes
3answers
387 views

How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?

I need some help with solving this integral involving Bessel function: $\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
53
votes
1answer
1k views

Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

Consider the following integral: $$\mathcal{I}(\mu,\nu)=\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$ where $J_\mu(x)$ is the Bessel function of the first kind: ...
34
votes
3answers
945 views

An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$

I need to calculate the following integral: $$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$ where $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$ ...