# Tagged Questions

Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.

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### Series expansion of elliptic integral involving n th order polynomial in the denominator

My goal is to find an expansion in powers of 1/ρ of the integral: $$I_n(\rho)=\int_\rho^{+\infty}\frac{dt}{(E_n(t))^2\sqrt{t^2-h_2^2}\sqrt{t^2-h_3^2}},\quad \rho \ge h_2$$ ...
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### Definite eliptic integral with trigonometric method

I'm Looking for integrals of the following kind $\int\limits_{0}^{M} \left( a +bx^2 + cx^3 \right)^{-\frac{1}{2}} \,dx$ Where $M$ is on of the roots of the polynomial. I used the trigonometric ...
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### Ellipsoid moment of inertia matrix

Some background info: torque $\tau$ is defined as $$\tau = I*d\omega$$ Where $I$ is the moment of inertia matrix and $d\omega$ is an object's rotational acceleration. As I understand it, the inertia ...
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### Does this three number mean have a name? (Carlson elliptic integrals)

Recently I found out about Carlson elliptic integrals, which have great symmetry properties and allow to compute every kind of elliptic integrals and other functions. The question is about the method ...
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### Point halfway around ellipse quadrant

I want to find the length between the centre of an ellipse and a point, P, on the ellipse, where the arc length between P and the intersection of the semi-minor axis with the ellipse is equal to the ...
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### Calculating X & Y coordinates of a point that is perpendicular to an ellipse point AND offset by -5

I am trying to calculate an offset point from a point on an ellipse - I need to be perpendicular to each point on the ellipse but 5 points in from the point on the ellipse. The result will probably ...
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### Evaluate $\int\frac{(-x_{0}+R\cos(\theta))\,d\theta}{\left((-x_{0}+R\cos(\theta))^2+(-R\sin(\theta))^2\right)^{3/2}}$

As the title suggests, how may I start evaluating the following integral: \begin{align}I=\displaystyle\int\dfrac{\left(-x_{0}+R\cos\left(\theta\right)\right)\,d\theta}{\left((-x_{0}+R\cos(\theta))^2+(...
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### Elliptic Integrals

In my homework I had to solve the following integral $\displaystyle\int_0^\pi \mathrm{d}\Psi \frac{\cos\Psi}{\sqrt{1+2s(1-\cos\Psi)}}$ with some constant $s\ll1$ The solution said this is an "...
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### The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page http://...
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### The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus ...
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### Integral Solution

During my attempt to solve the non-linear ODE $$m\ddot{x}+x-x^3=0$$ I have stumbled across the integral: \int{\frac{1}{\sqrt{\frac{1}{m}\left( \frac{x^4}{...
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### Differentiating a period of an elliptic curve under the integral sign

Let $$g = \frac{27J}{1 - J},$$ where $J$ is the absolute invariant, and define $$\Omega = \int_{\gamma(J)} \frac{dz}{\sqrt{4z^3 + g(z + 1)}}.$$ Here, $\gamma(J)$ is a contour in the complex plane that ...
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### How to compute $\int _0^{2\pi }\frac{1-\cos \left(t\right)}{\left(\frac{5}{4}-\cos \left(t\right)\right)^{\frac{3}{2}}}dt$

How to compute $$\int_{0}^{2\pi}\dfrac{1-\cos(t)}{\biggl(\dfrac{5}{4}-\cos(t)\biggr)^{\dfrac{3}{2}}} dt$$ I'm interested in more ways of computing this integral. My thoughts: I tired ...
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### Estimate ellipse segment length from total ellipse circumference

Assume you have the total length $S_{total}$ of the circumference of an ellipse in $\mathbb{R}^2$ with parameters $a$ and $b$. Is there an estimate relationship between $S_{total}$ and the length of ...
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### Question about specific arclength over an ellipse problem

to see an image of what I'm talking about click this link: https://i.gyazo.com/909ccf0113fd26d21797f411a756ba1e.png In this image, arclength A is what we desire to be calculated. Point P is given and ...
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### Definite integral with modified Bessel functions of first and second kind

I am interested in the following integral involving the modified Bessel functions of the first and second kinds of order one $I = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x$ For ...
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### Asymptotic behavior of elliptic integral (first kind)

I came accross some obstacles in proving that the time $T(\delta)$ taken by a pendulum to travel from $\theta=\pi-\delta$ to a considerably distant angle $\theta=\theta_0\in(0,\pi/4)$ diverges ...
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### The Divergence of The Elliptical Integral of First Kind $F(\phi,k)$

For what values of $k$ does the following elliptical integral of the first kind diverge? $$F(\phi,k)=\int\limits_0^{tan\phi} \frac{dt}{\sqrt{(1-t^2)(1-k'^2t^2)}}$$ where $\phi=\pi/4$ and $k'^2=1-k^2.$...
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### Evaluating Elliptic Integrals in terms of Gamma Function

Some complete elliptic integral of first and second kind $E(k)$ and $K(k)$ can be evaluated for some particular values of $k$ in terms of Euler Gamma function. For example, for $k = \sqrt{2}/2$, $E(k)$...
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### Series expression for $f(k)=4K(k)\left(\frac{b^2}{a^2-b^2}\right)-4E(k)\left(\frac{a^2}{a^2-b^2}\right)$

I'm trying to verify that the series expression for $f(k)=4K(k)\left(\frac{b^2}{a^2-b^2}\right)-4E(k)\left(\frac{a^2}{a^2-b^2}\right)$, where $a$ and $b$ are respectively the major and minor radii of ...
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