Elliptic integral Definition and 16 Threads

I'm looking at a proof of the asymptotic expression for the Elliptic function of the first kind https://math.stackexchange.com/questions/4064023/on-the-asymptotic-behavior-of-elliptic-integral-near-k-1 and I'm having trouble understanding this step in the proof: $$ \begin{align*} \frac{1}{2}...

I need to solve this integral which I suppose is an elliptic integral but don't know what kind, I'm not that familiar with them. Mathematica says that it can be expressed with elementary functions and gives the solution: ## -\frac{2\...

I have the first and second orders that I use in a magnetic simulator, but i need the thirth also to do also with magnetic cylinders accordingly paper: Do anybody have it in any code? I should pass to C++

I'm not sure if this should go in the homework forum or not, but here we go. Hello all, I've been trying to find a series representation for the elliptic integral of the first kind. From some "research", the power series for the complete form (## \varphi=\frac{\pi}{2} ## or ## x=1 ##) seems to...

Hello everyone. In the 3rd edition of Mechanics by Landau and Lifshitz, paragraph 14, there is a problem concerning spherical pendulum. Calculations leading to the integral $$ t=\int \frac {d \Theta} {\sqrt{\frac{2}{ml^2}[E-U_{ef}(\Theta)]}},$$ $$...

So the formula for an ellipse in polar coordinates is r(θ) = p/(1+εcos(θ)). By evaluating L = ∫r(θ) dθ on the complex plane on a circle of circumference ε on the centered at the origin I obtained the equation L = (2π)/√(1-ε^2). Why then does Wikipedia say that the formula for the perimeter is...

Hello I hope this is the right place to ask this question. For my thesis I need a way to invert a incomplete elliptic integral of the second kind. I believe the Jacobi elliptic functions are inverse of the elliptic integral of the first kind. The calculation I'm doing is symbolic so a...

How to prove: $\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $ where \[ k(\theta)=\sqrt\frac{4Rh \tan \theta}{Z^2+(R+h \tan \theta)^2}\] and $ K[k(\theta)] $ is the complete elliptic integral of the first kind...

Homework Statement Effectively, I'm trying to show the following two integrals are equivalent: \int_1^{1/k}[(x^2-1)(1-k^2x^2)]^{-1/2} dx = \int_0^1[(1-x^2)(1-(k')^2x^2)]^{-1/2}dx where k'^2 = 1-k^2 and 0 < k,k' < 1. Homework Equations One aspect of the problem I showed the following...

Hi all. I was reading this Wikipedia article: http://en.wikipedia.org/wiki/Risch_algorithm I have a couple of questions about \int \frac{x}{\sqrt{x^4 + 10x^2 - 96x - 71}} \, dx and \int \frac{x}{\sqrt{x^4 + 10x^2 - 96x - 72}} \, dx discussed in the article. How the heck did they get the...

Homework Statement Sub problem from a much larger HW problem: From previous steps we arrive at a complete elliptic integral of the second kind: E(k)=\int_0^{\pi/2} dx \sqrt{1+k^2\sin^2x} In the next part of the problem, I need to expand this integral and approximate it by truncating at the...

Homework Statement Given the differential equation u_{xx}+3u_{yy}-2u_{x}+24u_{y}+5u=0 use the substitution of dependent variable u=ve^{ \alpha x + \beta y} and a scaling change of variables y'= \gamma y to reduce the differential equation to v_{xx}+v_{yy}+cv=0Homework Equations I have no...

When replacing x with 1/kx then \int_{1/k}^\infty {\left[ {\left( {x^2 - 1} \right)\left( {k^2 x^2 - 1} \right)} \right]} ^{ - 1/2} dx = \int\limits_0^1 {\left[ {\left( {\frac{1}{{k^2 x^2 }} - 1} \right)\left( {\frac{1}{{x^2 }} - 1} \right)} \right]} ^{ - 1/2} \frac{{dx}}{{kx^2 }} I...

Homework Statement The problem is to calculate integral \int_{0}^{\pi/2}\frac{dx}{\sqrt{\sin{x}}} by transforming it into elliptical form (complete elliptical integral of first kind).

Hi, I was studying calculus and I had a problem while checking my results. I came to the following result: \int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin t}}\:\mathrm{d}t = \sqrt{2}\cdot \mathrm{F}\left(\frac{\pi}{4},\frac{1}{4}\right) \approx1.16817 However, Mathematica shows...

Taken from http://en.wikipedia.org/wiki/Elliptic_integral: Is it just me, or does it seem like there is an easier way to find the arc length of an ellipse? I thought elliptic integrals arose in giving the arc length of elliptic curves, which as far as I know are a lot different than ellipses.