# Tagged Questions

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### 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 ...
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### 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?
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### 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? ...
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### 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 $a$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### Incomplete gamma function in polar form

How can one write the incomplete gamma function in polar coordinates?
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### 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 : ...
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### 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 ...
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### 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 ...
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### 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.
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### 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: $$I=\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi$$ For $a = 0$, the integral is easily ...
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### $\int_{-1}^{1} x^{k+i} P_n(x)dx$, $P_n$ 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 ...
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### 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 ...