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

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Compute the asymptotic expansion of the integral by Watson's Lemma

Use Watson's Lemma to find the asymptotic expansion of the following integral as $\lambda \to \infty$ with $\lambda>0.$ Assuming that $\phi (t)$ is infinitely differentiable on $[0,1].$ ...
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1answer
46 views

Showing that the complete elliptic integral of the second kind can be represented by a particular series

I'm trying to show that the complete elliptic integral of the second kind can be represented as: $$E(k)=\frac{\pi}2 ...
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32 views

Proving relationship between major and minor radius of ellipse of constant radius

So, I've got a real doozy of a question. I'm trying to provide a proof for the relationship between the major and minor radius ($a$ and $b$, respectively) of an ellipse of constant circumference as ...
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2answers
27 views

Asymptotic expansion of the complete elliptic integral of the first kind

The complete elliptic integral of the first kind is defined as $$K(k) = \int_0^{\pi/2} \frac{d x}{\sqrt{1 - k^2 \sin^2 x}}.$$ I would like to derive (at least the first term of) the asymptotic ...
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2answers
55 views

Elliptic integral evaluation

How to integrate ( $ r_o, r_b$ constants) $$ \int \sqrt{\dfrac{r_o^2- r^2}{r^2-r_b^2}} \, dr, (r_o > r > r_b > 0)\, ? $$ With Mathematica got its coefficient imaginary, needing to take ...
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1answer
77 views

How does a simple elliptic integral solve this monster?

During some electromagnetics calculation regarding a loop antenna I stumbled across the following integral $$\int_0^{\pi/2} \frac{d\phi}{\big(1+\frac{k}{k-2}\cos(2\phi)\big)^{3/2}}$$ and Mathematica ...
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2answers
94 views

Approximation for elliptic integral of second kind

My (physics) book gives the following approximation: $\int_{-\pi/2}^{\pi/2} \sqrt{1-(1-a^2) \sin(k)^2} dk \approx 2 + (a_1 - b_1 \ln a^2) a^2 + O(a^2 \ln a^2)$ where a1 and b1 are "(unspecified) ...
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33 views

Reference about elliptic integral and Jacobi Inversion Problem

I read the section about Abel's theorem and the Jacobi Inversion Problem on the book of Forster, "Lectures on Riemann Surfaces". I would like if there were some books which treats more in detail this ...
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1answer
65 views

An elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system

Arnold in his essay On teaching mathematics made the following statement: The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only ...
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1answer
55 views

Complete Elliptical Integral for the parameters greater than 1 or less than -1

I am trying to compute the Complete elliptical integral of second kind kind in Mathematica with Parameter m=-19.7 .Following is the response from Mathematica. Input:EllipticE[-19.71] Output:4.81841 ...
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47 views

Reducing integral

let $$I=\int \frac{dx}{\sqrt{mx^3-x^2+n}}$$ How do we reduce $I$ to an elliptic Integral of the first Kind ? where $m,n>0$ are constants.
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139 views

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} ...
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36 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
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1answer
19 views

integral of a function bounded over an elliptic area

I've been stuck with the following integral, I know I have to use substitution, but I don't know how. Let $S=\{(x,y):\frac{x^2}{a^2}+\frac{y^2}{b^2}\leq 1\}$. Show that $$\int \int_{S} \left( ...
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33 views

Calculate volume in ${\mathbb{R}}^{3}$ bounded by the given function, inside the region $S$.

Let $S=\{(x,y): \frac{x^2}{a^2}+\frac{y^2}{b^2}\leq 1\}$. Verify that: $$\iint\limits_S {\left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}}\right ...
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1answer
92 views

Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
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1answer
184 views

Is this integral reducible to an elliptic integral?

I believe if $k=0$ the following integral is reducible to an elliptic integral. If $k > 0$ is it possible to reduce it to an elliptic integral or some other special function? $$\int_\rho^x \sqrt{1 ...
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33 views

Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
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Further integrals of the seemingly unintegrable

I remember having to solve the following problem: Let \begin{equation} I_n=\int_0^{1}\frac{x^ndx}{\sqrt{x^3+1}}. \end{equation} Prove \begin{equation} ...
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1answer
61 views

Double Integral Query

Looking for advice/direction on the following query please. Ok, here is the problem: If we have $$ \vec{T} = \frac{\mu IbN}{4\pi L}\int_{-L/2}^{L/2}C\int_{0}^{2\pi}\frac{\cos(\theta)}{p^3}d\theta ...
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Is the elliptic integral bounded?

Is the elliptic Integral of the first kind bounded (also for complex argument)?
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Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
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1answer
240 views

Fourier series of $\sqrt{1 - k^2 \sin^2{t}}$

I'm struggling with a Fourier series. I need to find the Fourier series of the following function. That's the function under study: $f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$. The function ...
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1answer
103 views

Indefinite integral typo in Gradshteń≠n: reciprocal square-root of sixth degree polynomial

The indefinite integral below, quoted from Gradshteń≠n's Table of Integrals, Series, and Products, 7th ed., (bottom of p.104) appears to contain at least two typos (highlighted in purple). ...
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Integrating the function $\alpha^{2}\sin^{2}\left( \dfrac {2 \pi x}{w}\right)$

I need to find $\int ^{x}_{0} \alpha^{2}\sin^{2}\left( \dfrac {2 \pi x}{w}\right) dL$ where $L=\dfrac {2kE\left( \dfrac {2\pi \alpha }{k}\right) }{\pi}\approx \dfrac {k+w}{2}\left( \dfrac {3\left( ...
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3answers
257 views

Integral of inverse of square root of quartic function with real roots

I was doing a physics problem and in order to finish it, I need to prove that: $\int_{x1}^{x2}\frac{dx}{{((x1 - x)(x - x2)(x - x3)(x - x4))}^{1/2}} = \int_{x3}^{x4}\frac{dx}{{((x1 - x)(x - x2)(x - ...
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3answers
204 views

How do I integrate $\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}$

How do I evaluate this indefinite integral, for $|k| < 1$: $$ \int\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}\mathrm{d}x $$ I tried the change of variable $t=\sin x$, and obtained two integrals, but I ...
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Is there a nice way to solve this Ramanujan-like approximation?

I have managed to approximate $l=\frac{2 \sqrt{4 \pi ^2 A^2+W^2} E(\epsilon )}{\pi }$ with parameter $\epsilon =k=\frac{2 \pi A}{\sqrt{4 \pi ^2 A^2+W^2}}$ alike Ramanujan as $l \approx ...
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1answer
64 views

Changes of variables to get an Elliptic Integral of the First Kind

I'm working with a non-linear second order ODE which has an analytical solution in terms of the Jacobi elliptical function $sn(u|k^2)$. The equation is $y''=y(\gamma - \frac{y^2}{2})$ where $\gamma$ ...
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Approximating an Infinite Summation into Closed Form

I want to be able to approximate the following in closed form $$L=\sqrt{\gamma ^2+4 \pi ^2 \psi ^2} \sum _{j=0}^{\infty } \frac{\left(\frac{(2 j-1)\text{!!} (2 \pi \psi )^j}{(2 j)\text{!!} ...
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How can I find my eccentricity (k, of the incomplete elliptic integral of the second kind) using a binomial series or root-finding algorithm

My main objective is to rearrange the following to find $A=$. I start with $C=\int^{B}_{0}\sqrt {1+\dfrac {A^{2}}{B^{2}}\cos^{2}\left(\dfrac {x}{B}\right) }dx$ By substituting $y=x/B$ and using ...
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Value of an elliptic integral of the first kind

The elliptic integral of the first kind $$ \int_0^{\pi/2}{\frac{du}{\sqrt{1-k^2\sin^2{u}}}} $$ cannot be expressed in terms of standard functions. But in the following context from The Pendulum by ...
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2answers
43 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
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2answers
138 views

An identity of an Elliptical Integral

Suppose $0<k<1$ and $\displaystyle K(k)=\int_0^1\frac{\mathrm{d}x}{\sqrt{(1-x^2)(1-k^2x^2)}}$. Let $\tilde{k}$ be $\tilde{k}^2=1-k^2$. Show that $$\displaystyle ...
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1answer
97 views

Elliptic integral $\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk$

Question: Prove that $$\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk=\frac{1}{16\pi}\Gamma^4\left( \frac{1}{4}\right)$$ My attempt Start by the transformation $$k \to \frac{2\sqrt{k}}{1+k}$$ ...
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1answer
49 views

Numerical evaluation of the first (K) and second (E) complete elliptic integrals

To get a numerical evaluation of the first (K) and second (E) complete elliptic integrals: $$K(k)=\int_0^1\frac{dt}{(1-t^2)^{1/2}(1-k^2t^2)^{1/2}}, \ \ \ \ \ ...
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1answer
33 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
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153 views

Elliptical Integrals

I was trying to figure out the length of the arc in a single cycle of a sinusoidal curve and I used the curve length formula to arrive at $$\int_0^{2\pi}\sqrt{1+\cos^2x}\ dx,$$ which I am fairly ...
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Expression of $2\int_{0}^{\frac{1}{r_{0}}}\frac{du}{\sqrt{\frac{r_{0}-r_{s}}{r_{0}^{3}}-u^{2}\left(1-u r_{s}\right)}}$ in terms of elliptic integrals

In Gravitation by Misner et al. 1973 the authors state that (calculus related to the Schwarzschild metric page 678) : ...
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1answer
71 views

Closed form of $\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1)$

Is there a known closed form of the series below? $$\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1]$$
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1answer
72 views

Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
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1answer
125 views

TaylorSeries of complete elliptic integral of the first kind

I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$. Can someone help me? $$ K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 sin^2t}} $$ Results so far: $$ ...
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212 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\bigl(\frac{1}{e^n}\bigr)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
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5answers
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1answer
177 views

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: \begin{equation} I=\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi \end{equation} For $a = 0$, the integral is easily ...
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1answer
69 views

Infinite series representation of elliptic integrals $F(k,\phi)$ and $E(k,\phi).$

Here is my working. For a natural number $n,$$$2^{2n-1}\left(-1\right)^n\sin^{2n}\phi=\tfrac{1}{2}\left(-1\right)^n\binom{2n}{n}+\sum\limits_{r=0}^{n-1}\binom{2n}{r}\left(-1\right)^r\cos ...
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4answers
237 views

Prove that these two curves have the same length

My midterms are approaching, and I was going through some of our past Calculus midterms when I stumbled upon this question from 1996: Show that these two curves, $$(\Gamma) : \frac ...
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1answer
65 views

How to get arc-length of polar function $r= 4(1-\sin{\phi})$?

How can I get arc-length of this polar function? $$ r= 4(1-\sin{\phi})$$ $$-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$$ I know that arc-length of polar function can get calculate by ...
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25 views

Relationship between elliptic integrals of first and second order.

A colleague and I are making an issue where order to reach the result we are asked to prove identity between elliptic integrals first and second order. This is: $$\frac{V}{\pi}\left( ...