Questions tagged [elliptic-integrals]

Question 1

I found this problem from a FB page $$\Omega = \int\limits_3^\infty {\frac{{{x^4}}}{{\sqrt {{{\left( {{x^4} + 2{x^2} + 4} \right)}^3}} }}dx}$$ Here what I tried to find the closed form: $${\text{We ...

Question 2

Consider the complete elliptic integrals $K, E$ which are defined as follows: $$K(m)=\int_{0}^{\pi/2}\frac{dx}{\sqrt{1 - m\sin^{2}x}},\,E(m)=\int_{0}^{\pi/2}\sqrt{1-m\sin^{2}x}\,dx$$ I want to know ...

Question 3

Let $a\ge0$, show that: $$ \int\limits_0^1\frac{a+2x}{\sqrt{x(1-x)(a+x)(1+a+x)}}dx = \pi \tag{1} $$ For $a=0$, the integral is: $\displaystyle\int_0^1\frac{dx}{\sqrt{1-x^2}}=\frac{\pi}{2}$ Splitting ...

Question 4

In the process of attempting to find a travelling soliton solution to the massive Thirring model, I have encountered the below system of ODEs, the solution to which I do not know how to even begin. I ...

Question 5

While reading the solution of the Watson's triple integral, I stumbled upon the following identity: For $0 \leq \alpha \leq \frac{\pi}{2}$, we have $$ \int_{0}^{\frac{\pi}{2}}K(\sin(\theta)\sin(\alpha)...

Question 6

I am interested in evaluating the following improper integral analytically $$ F=\int_0^\infty \frac{\sinh(ax)}{\sinh(\pi x)}\, P_{ix-\tfrac{1}{2}}(b)\,\mathrm{d}x \,, $$ with $a\in[-\pi,\pi]$ and $b\...

Question 7

Context Being: $$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2{t}}}=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(2n)!^2k^{2n}}{2^{4n}n!^4},\hspace{.5cm}k \in(0,1)\tag{1}$$ and: $$E(k)=\int_{0}^{\pi/2}{\...

Question 8

I have been reading Zaid Alyafeai's book Advanced Integration Techniques, and I came across the following identity: $$K(k)=\frac1{1+k}K\left(\frac{2\sqrt k}{1+k}\right),$$ where $ K(k)$ is the ...

Question 9

Someone asked me a question about integrals. $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}x \csc^2x\sqrt{4-\csc^4x}dx$$ I broke it down into several integrals using integration by parts. enter image ...

Question 10

Consider the polynomial $P(x)$ of degree $4$, and assume we have real roots $a>b>c>d$. I want to study the quantities \begin{align*} \int_b^a\frac{1}{\sqrt{P(x)}}dx, \quad \int_d^c\frac{1}{\...

Question 11

I'm working on an architectural project with some vaults and elliptical arches. I've arrived at a system of equations with three unknowns, but one of these equations is an elliptical integral of ...

Question 12

I'm not all too familiar with the theory of elliptic integrals but I wanted to know if the integral $$\int_{0}^{1}\frac{\mathrm{dt}}{\sqrt{n-t^3}},\quad 1 \le n \in \mathbb{Z}$$ ever has a closed form ...

Question 13

Let $K$ be the complete elliptic integral of the first kind: $$K(k)=\int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}$$ and let $k'=\sqrt{1-k^2}$. Then the quantity $q=e^{-\pi K(k')/K(k)}$, also called the ...

Question 14

I've been trying to work on the following integral, $$I(n,m,a)=\int_0^1 x^{n-1}(1-ax)^{m-1}K(x)\,dx\,\,\, | \,\,\, n,m \in \mathbb{Q}^{+}, a\in (0,1)$$ Where, $K(x)$ is complete elliptic integral of ...

Question 15

This might be a simple question, but I'm quite new to algebraic geometry. I know we can relate the genus $2$ sextic curve $$ W_6=\left\{(x,y)\in \mathbb{C}\, : \, y^2=1-x^6\right\}$$ To the genus $1$ ...

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Questions tagged [elliptic-integrals]

Closed form of $\Omega = \int\limits_3^\infty {\frac{{{x^4}}}{{\sqrt {{{\left( {{x^4} + 2{x^2} + 4} \right)}^3}} }}dx} $

For what integer $n$ does $E(2/n)/K(2/n)$ reduce to Gamma functions?

On $\int_0^1\frac{a+2x}{\sqrt{x(1-x)(a+x)(1+a+x)}}dx=\pi$

Elliptic solution to system of ODEs resulting from travelling soliton solution of the massive Thirring model

Proving $\int_{0}^{\frac{\pi}{2}}K(\sin(\theta)\sin(\alpha)) \, \mathrm{d}\theta=K(\sin(\alpha/2))^2$

Simplification of an analytically evaluated integral expressed in terms of elliptic integrals

About Legendre's like relations generalizations.

Elliptic integral. Proof, without quadratic transformation, of $K(k)=\frac1{1+k}K\left(\frac{2\sqrt k}{1+k}\right)$?

Help in the elliptic integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}x \csc^2x\sqrt{4-\csc^4x}dx$.

Proving two elliptic integrals over two different set of roots of the same polynomial are the same

Ecuation system with Incomplete Elliptic Integral of Second Kind [closed]

Does $\int_{0}^{1}\frac{\mathrm{dt}}{\sqrt{n-t^3}}$ ever have a closed form?

On $q=z+2z^5+15z^9+150z^{13}+\cdots$

Generalizing $\int_0^1 x^{n-1}(1-ax)^{m-1}K(x)\,dx$

Morphism from sextic curve to elliptic one

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