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

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12
votes
0answers
70 views

Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
1
vote
0answers
26 views

Jacobi modulus and Weierstrass $\wp$

Let $0 < k < 1$ and $$K := \int_0^{\pi/2} \frac{1}{\sqrt{1 - k^2 \sin^2(\theta)}} \, \mathrm{d}\theta, \; \; K' := \int_0^{\pi/2} \frac{1}{\sqrt{1 - (1-k^2) \sin^2(\theta)}} \, ...
3
votes
1answer
40 views

Elliptic integral equality

I would like to know how to prove following integral equality : $$\int_{0}^{\frac{\pi}{2}}\frac{1-k^2}{(1-k^2\sin^2\theta)^\frac{3}{2}}d\theta = ...
2
votes
1answer
42 views

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 ...
2
votes
0answers
26 views

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 ...
0
votes
0answers
17 views

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 ...
0
votes
0answers
8 views

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 ...
1
vote
1answer
65 views

Definite integral with trigononmetric functions

I have arrived at definite integral with trigonometric functions $I(a, b) = \int_0^{2 \pi} \frac{1 - a \sin(\theta) - b \cos(\theta)}{(1 +a^2 +b^2 - 2 a \sin(\theta) - 2 b \cos(\theta) )^{3/2}} ...
0
votes
2answers
37 views

EllipticF problem with Maple

When I solve for EllipticF[1.4, 0.9] with Mathematica or ellipticF(1.4, 0.9) with Mupad I get ...
2
votes
0answers
52 views

How to evaluate following integral?

Suppose the integral $$ \tag 1 I = \int \limits_{-\pi}^{\pi}dx \int \limits_{-\pi}^{\pi}\frac{dy}{\tau - \cos (2x) -2\cos(x)\cos(y)}, \quad t > 3 $$ How to evaluate it in terms of elliptic ...
1
vote
0answers
35 views

How to make analytic continuation and compute imaginary part

Suppose I have the function $$ \tag 0 G(x) = g(x)K\left(k(x)\right), $$ where $$ k(x) = \frac{4\sqrt{x}}{(x-1)^{\frac{3}{2}}(x+3)^{\frac{1}{2}}}, $$ $$ g(x) = \frac{2}{\sqrt{x}}k(x), $$ and $K(x)$ is ...
5
votes
2answers
329 views

Find the ratio of integrals $\int_0^1 (1\pm x^4)^{-1/2}\,dx$

How to find this ratio $$\frac{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1+x^{4}}}\mathrm{d}x}{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1-x^{4}}}\mathrm{d}x}$$ without evaluating each integral? ...
5
votes
1answer
104 views

An elliptic integral?

I ran into an integral a little while ago that looks like an elliptic integral of the first kind, however I am having trouble seeing how it can be put into the standard form. I've tried messing ...
5
votes
1answer
101 views

May I know how this integral was evaluated by using the theory of elliptic integrals?

I can not solve the following integral using the theory of elliptic integrals: $$\int_a^b \frac{\sin(x)}{\sqrt{c-\sin(x)}}dx$$ Where $a\geq 0, b>0, c>0$. Wolfram$|$Alpha showed the following ...
0
votes
1answer
77 views

Computing the period via elliptical integral

I am having trouble showing that the period (T) of this system can be expressed in terms of an elliptical integral. Given the dynamical system governed by the differential equation below: ...
0
votes
1answer
29 views
2
votes
2answers
116 views

Solving $\int_0^{\pi/2} \sin^2\theta \sqrt{1-k^2\sin^2 \theta}d\theta $

I have the following integration to solve. $$f(k) = \int_0^{\pi/2} \sin^2\theta \sqrt{1-k^2\sin^2 \theta}d\theta,\quad0<k<1$$ assuming $\sin\theta = t$ which results $d\theta = ...
1
vote
1answer
49 views

Numerically solving the equation of a simple pendulum with Runge-Kutta.

I am trying to solve the equation $\dfrac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \dfrac{g}{L} \sin{\theta} = 0$ using Runge-Kutta. I have alread split it into the following equations ...
3
votes
1answer
57 views

Elliptical Integral that diverges at one point

I have to solve the following integral $$I=\int_{\lambda_1}^yd\lambda\frac{1}{1-\lambda}\sqrt{\frac{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_4)}{\lambda-\lambda_3}}$$ where ...
0
votes
0answers
29 views

can this functional of the Hardy Z function be written as an elliptic theta function?

Can H(t), Equation 33 of http://vixra.org/pdf/1510.0475v7.pdf be expressed as an elliptic theta or related function ? $H(t)= {\frac {4\,i\zeta \left( -i/2 \left( i-2\,t \right) \right) \pi \, ...
1
vote
3answers
104 views

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 ...
7
votes
1answer
150 views

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 ...
14
votes
2answers
186 views

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 ...
0
votes
0answers
44 views

Integral Solution

During my attempt to solve the non-linear ODE \begin{equation} m\ddot{x}+x-x^3=0 \end{equation} I have stumbled across the integral: \begin{equation} \int{\frac{1}{\sqrt{\frac{1}{m}\left( ...
1
vote
0answers
104 views

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 ...
0
votes
0answers
60 views

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 ...
0
votes
0answers
20 views

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 ...
1
vote
1answer
57 views

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 ...
5
votes
0answers
76 views

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 ...
2
votes
1answer
95 views

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 ...
0
votes
0answers
38 views

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 ...
4
votes
0answers
69 views

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 $k = \sqrt{2}/2$, ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
37 views

Arc Length for Superposition of Sinusoidal Curves

I am wanting to compute the arc length, $s$, of a superposition of two sinusoidal functions--say $$y(x) = A\cos\left(k_1 x\right)+B\cos\left(k_2 x\right).$$ There is a special relationship between ...
11
votes
1answer
103 views

Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$

Let denote $K$ and $E$ the complete elliptic integral of the first and second kind. The integrand $K(\sqrt{k})$ and $E(\sqrt{k})$ has a closed-form antiderivative in term of $K(\sqrt{k})$ and ...
3
votes
1answer
97 views

Improper integral of $\sqrt{x^4 + 1} - x^2$ [duplicate]

I'm having a little trouble with this integral: $\int^\infty_0 (\sqrt{x^4+1} - x^2)\,dx$. Using the likes of Maple, I can easily find that it takes the form $-\frac{2}{3}\sqrt{2}(1+i)K(i) - ...
1
vote
2answers
159 views

map the UHP to an equilateral triangle

Explain how the upper half-plane can be mapped one-to-one and conformally onto an equilateral triangle. Thanks,
0
votes
2answers
85 views

Second order nonlinear differential equation $x''+Hx =A(1-J/(2x^2))$

I have arrived at a differential equation and I need to solve for $x$. $$ \frac{\mathrm{d}^2x}{\mathrm{d}E^2}+Hx =A\left(1-\frac{J}{2x^2}\right) $$ where $H$, $A$, and $J$ are constants. I know ...
3
votes
0answers
148 views

Evaluate $\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$

We have the following integral: $$\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$$ where $P_3(t)$ is a third-degree polynomial with all coefficients different from zero. Is it an elliptic integral? ...
4
votes
2answers
173 views

Surface area of the ellipsoid $\frac{x^2}{16}+\frac{y^2}{8}+z^2=1$

My professor gave us this question on a calculus II quiz. One of my calculus III pals suggested I use surface integrals, but that tool is not available to us (I don't know how to use it yet, nor do my ...
3
votes
3answers
88 views

Complete elliptic integral of the first kind $K(m)$ asymptotc expansion at $m = -\infty$

What is the asymptotic behavior of $K\left(-\frac{1}{\delta^2}\right), \delta > 0$ when $\delta$ tends to zero? Here $$ K(m) = \int\limits_0^{\pi/2} \frac{d\theta}{\sqrt{1 - m\sin^2 \theta}}, $$ ...
0
votes
0answers
123 views

Handling complex arguments of elliptic integrals in Maple

I want to solve the integral \begin{equation} I(y)=\int_0^y\sqrt{\dfrac{x(1+ax)}{(1+ax)^2-b^2}}\,\mathrm{d}x,\qquad a<0,\; b\in(0,1),\; y>0,\; (1+ax)^2-b^2>0, \end{equation} using Maple 18. ...
2
votes
1answer
39 views

Is this function defined in terms of elliptic $\mathrm{K}$ integrals even?

Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} ...
0
votes
1answer
81 views

Good books about elliptic integralsa, hypergeometric and special functions

Can you please tell me some good books from where I can learn elliptic integrals and special functions like hypergeometric functions?
0
votes
2answers
61 views

How to evaluate the length of the perimeter of a low eccentricity ellipse?

Given that $ e= \frac{a^2-b^2}{b^2} $ , and $L$ is the length of the perimeter, which equals $4aE(e, \pi/2)$, find the length of the perimeter up to $e^2$ in terms of $a$ and $b$. How does one begin ...
3
votes
1answer
122 views

Closed form for an almost-elliptic integral

Does $$\int_0^{2 \pi} \log\left(\frac{1}{2}[1+\sqrt{1-(a \sin\phi)^2}]\right) d\phi $$ have a closed form ? An approximation for small $a$ is $2E-\pi$, but it is the exact form that is needed ...
7
votes
1answer
223 views

Integral relations in Fricke and Klein

Can someone please explain how Fricke and Klein obtain the integral relationa stated at the top of p. 34 in this book? The entire book can be previewed on Google Books. It is an old book and I do not ...
2
votes
1answer
69 views

Limit evaluation with elliptic integrals

Prove the following involving elliptic integrals: $$ \lim_{u\to 0 } \dfrac{K(u)- E(u) } {1 - \sqrt {1-u}} = \frac{\pi}{2} $$
2
votes
1answer
60 views

Evaluating $\int^{x_2} _{x_1} \sqrt{a - b x^m} ~dx $

Is there any way to evaluate $$\int^{x_2} _{x_1} \sqrt{(a - b x^m)}~ dx $$ where $x_{12} = \pm (a/b)^{1/m}$ without elliptic functions or hypergeometry? Or just any way to solve it. My attempt is to ...
1
vote
1answer
77 views

Can an integral of a function that is not well behaved be finite?

Consider the following integral which gives the time period of simple pendulum where $\theta_0$ is the initial inclination of pendulum with vertical. ...