1
vote
0answers
25 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta$, where a and b are real numbers. If we consider ...
0
votes
0answers
44 views

Integral of Hypergeometric Function with polynomial and exponential

I was working on some mathematical derivations and faced this integral: $$I=\int_{0}^{\infty}x^{\alpha-1}e^{-\beta x}{_2F_1}{(a,b;c;1-hx)}\,\mathrm{d}x$$ how can I integrate it?
14
votes
1answer
231 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
0answers
42 views

Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$\int ^1 _0 dx \frac{\ln(x-a)}{x-b}$$ by turning it into something in terms of dilogarithms. But for certain values of $a$ ...
3
votes
1answer
59 views

What does this paper mean by “$f(x)$ is practically a rational function”?

The paper "Infinite integrals involving Bessel functions by contour integration" by Qiong-Gui Lin gives a method to solve integrals of the form $\intop_{0}^{\infty}x^{v}f(x)J_{v}(qx)\, dx$. One of the ...
2
votes
2answers
32 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
3
votes
1answer
48 views

Does $\int { y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$ have a closed form

I am trying to solve the following indefinite integral $$F_Y(y) = \int {y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$$ Where $J_0$ is the Bessel function of the first kind. I tried ...
0
votes
0answers
40 views

Definite integral with doable improper case

Is there a way to evaluate one or both of the following integrals: $$ \int_{a_1}^{a_2} e^{ib_1(x+b_2 \sqrt{1+x^2})}dx \quad \text{and}\quad \int_{a_1}^{a_2}\frac{x}{\sqrt{1+x^2}} e^{ib_1(x+b_2 ...
1
vote
1answer
95 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 ...
8
votes
2answers
232 views

Difficult infinite integral involving a Gaussian, Bessel function and complex singularities

I've come across the following integral in my work on flux noise in SQUIDs. $$\intop_{0}^{\infty}dk\, e^{-ak^{2}}J_{0}\left(bk\right)\frac{k^{3}}{c^{2}+k^{4}} $$ Where $a$,$b$,$c$ are all positive. ...
1
vote
0answers
55 views

Analytical evaluation of integral

I would like to evaluate the following integral analytically, but Mathematica does not give me an answer: $$ \int_0^1 dr \ e^{(1-2r)x^2} \left[p(r,x) Y_0\left(2x^2\sqrt{r-r^2}\right)+q(r,x) ...
4
votes
3answers
87 views

Proving $\int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1$.

By testing in maple I found that $$ \int_0^1 \frac{\mathrm{d}x}{1-\lfloor \log_2(1-x)\rfloor} = 2 \log 2 - 1 $$ Does there exists a proof for this? I tried rewriting it as an series but no luck ...
4
votes
2answers
61 views

2D Integral of Bessel Function and Gaussians

I've run into the following integral, and I'm not sure how to evaluate it. $$F(k)=\int ...
1
vote
2answers
88 views

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
3
votes
1answer
30 views

Calculate integral with $\Gamma$ and $B$

The integration is like: $$\int_{a}^{b}\left(\frac{b-x}{x-a}\right)^{p}dx$$ with $0<p<1$ Answer is $(b-a)p \frac{\pi}{\sin p\pi}$ Apparently, we can reversely construct $$\Gamma(1-p) ...
2
votes
2answers
127 views

Compute the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$ or $I=\int_{d}^1 y^{-a}(1−y)^{b-1} dy$

I need to calculate the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$, where $a$, $b$ are REAL NUMBERS and $b>0$. (my goal is to determine the definite integral $I=\int_{d}^1 y^{-a}(1−y)^{b-1} ...
5
votes
1answer
130 views

Prove that $\exists\,! \,\lambda \in (1/5,1/4)$ such that $\frac{1}{2\pi}\int_0^{2\pi}e^{\sin x}\,\mathrm{d}x=e^{\lambda}$

The following question came up in chat Prove that $\exists\,! \,\lambda \in (1/5,1/4)$ such that $\displaystyle\frac{1}{2\pi}\int_0^{2\pi}e^{\sin x}\,\mathrm{d}x=e^{\lambda}$ Now the integral ...
1
vote
1answer
49 views

$\int xtanx$ and the Clausen Function

I have been attempting to evaluate $\int x \tan x \;\mathrm{d} x$. My first instinct was integration by parts, which produces $-x \ln|\cos x|+\int \ln|\cos x| \;\mathrm{d} x$. I have read online ...
0
votes
0answers
46 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 ...
0
votes
0answers
12 views

closed form of integral of special function? $\int_{0}^{1} e^{a\, q^{1/k}\, {}_1 F_{2}(1,1/k,1 + 1/k;q)} d q$

Take the following integral, defined by hypergeometric functions: $$\int_{0}^{1} e^{a\, q^{1/k}\, {}_1 F_{2}(1,1/k,1 + 1/k;q)}d q$$ (there is a similar formulation Lerch). I think the series ...
1
vote
1answer
74 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
2
votes
0answers
32 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
0
votes
0answers
22 views

Integrating products of Hankel and Riccati Bessel functions

I want to do the integral: $$ \int_0^\infty dr h_l^+(kr)\hat j_l(kr) $$ where $h_l^+$ is the type 1 Hankel function, $\hat j_l$ is the type 1 Riccati-Bessel function. I would like a algebraic ...
3
votes
1answer
123 views

How to reproduce the Mathematica solution for $\int(\cos x)^{\frac23}dx$?

I entered this integration problem to Mathematica Online Integrator an got a solution I would never have been able to find manually. $$\int\root 3 \of{\cos(x)^2}\,dx=\frac{(-3\cos(x)\root 3 ...
4
votes
1answer
69 views

Elliptic integral $\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk$

Question: Prove that $$\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk=\frac{1}{16\pi}\Gamma^4\left( \frac{1}{4}\right)$$ My attempt Start by the transformation $$k \to \frac{2\sqrt{k}}{1+k}$$ ...
10
votes
1answer
143 views

Integral $\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3$

Hi I am trying to prove this below. $$ I:=\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3 $$ where $$ \zeta_3=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I am ...
1
vote
1answer
82 views

Bessel function and upper bound

I'm stuck on this following problem: Let $G$ a function such that $0\leq G(t)\leq 1$, and $G(t)=1$ if $B^2\leq t\leq 4B^2$, with $\operatorname{supp}G\subset [\frac{1}{4}B^2, 9B^2]$ and $G^{(j)}\ll ...
5
votes
3answers
119 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
1
vote
1answer
47 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 ...
2
votes
1answer
94 views

Integral involving the Spherical Bessel Function of the First Kind

How can I prove the equation below using Spherical Bessel Function Recurrence Relation? (where $ j_{n}(x) $ means Spherical Bessel function of first kind) Definition using BesselJ function: $$ ...
3
votes
0answers
30 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 ...
3
votes
2answers
199 views

Evaluation of another definite integral

I have a definite integral that I am trying to solve. Any hint or reference is urgently sought. , where $r$ is any positive integer while $\psi$ and $\nu$ are positive real numbers.
0
votes
0answers
23 views

Incomplete gamma function in polar form

How can one write the incomplete gamma function in polar coordinates?
0
votes
0answers
29 views

The representation of the Gaussian Q-function

The gaussian Q-function is well-known to be given in two famous forms, given as: $$ \mathcal{Q} = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty}\! \operatorname{exp}\left\{ -\frac{y^2}{2} \right\} ...
5
votes
2answers
109 views

Integral $\int_0^1 \frac{x\log x+1-x}{x \log^2 x}\log(1+x)\, dx=\log\frac{4}{\pi}$

Hi I am trying to prove this $$ I:=\int_0^1 \frac{x\log x+1-x}{x \log^2 x}\log(1+x)\, dx=\log\frac{4}{\pi}. $$ Thanks. This is just a beautiful integral for many reasons. Logs are everywhere and an ...
7
votes
4answers
162 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 ...
4
votes
2answers
157 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} ...
4
votes
2answers
162 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 : ...
2
votes
0answers
102 views

An Integral possibly related to Legendre polynomials

Consider the integral $$\int_0^1\frac{(t^2-1)^a}{(t-u)^{b+1}}dz$$ where $b\gg a$, with $a,b$ integers and $u>1$. I know you can write this integral as the sum of two hypergeometric functions but ...
1
vote
1answer
41 views

What is the integral containing decaying exponential function?

I am trying to figure out properties of the following integral: $$p(t)=\int_{0}^{t} e^{\alpha(t-t')} f(t')dt', \hspace{1 cm} t>t'$$ I would google and read more info about this integral but I do ...
1
vote
1answer
28 views

Incomplete Gamma function as Meijer G

How can I write the incomplete Gamma function in terms of the Meijer-G function? Assume that the Incomplete Gamma is given as: where alpha, beta and m are real positive constants.
4
votes
1answer
123 views

Does $\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi $ have a closed form in terms of elliptic integrals?

Consider the following integral for real $a, b$ such that the square root is real: \begin{equation} I=\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi \end{equation} For $a = 0$, the integral is easily ...
1
vote
1answer
58 views

$\int_{-1}^{1} x^{k+i} P_n(x)dx$, $P_n$ Legendre polynomial.

I was wondering whether there is a way to say what $$\int_{-1}^{1} x^{k} P_n(x)dx$$ is, where $k,n$ are positive integers or zero and $P_n$ is the n-th Legendre polynomial? I am looking for an ...
3
votes
1answer
40 views

Properties of a Mehler's type integral

When computing the resolvent of the Laplace beltrami opetator on $S^n$ for even dimension, $n=2k$, I came across the following integral $$ ...
6
votes
0answers
226 views

Integral $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful (yes Beautiful) and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 ...
14
votes
1answer
564 views

Integral$=-\frac{4}{3}\log^3 2-\frac{\pi^2}{3}\log 2+\frac{5}{2}\zeta(3)$

Hi I have been trying to prove this $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
2
votes
1answer
81 views

Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems its a 'simple' property, but I haven't been able to ...
3
votes
1answer
92 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
1answer
66 views

Integrate square root of 4th grad polynomials

During some calculations for a program I came upon this Integral which I am not able to solve. I already tried Matlab but it didn't help me. Here is the Integral: $$\int\left(\sqrt{\sum_{0}^{5} 9 ...
2
votes
0answers
172 views

The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

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 training ...