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

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2
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0answers
14 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
2answers
41 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
57 views

the complete set of half-period formulas in terms euler's formula

Given the elliptic modulus $k$ ,such that the complementary modulus is defined by $$k'\equiv \sqrt{1-k^2}$$,the jacobi amplitude $$\phi\equiv am(u|k)$$ and $K(k)$, is the complete elliptic integral of ...
0
votes
0answers
25 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 ...
3
votes
0answers
31 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
11 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
26 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 ...
9
votes
1answer
80 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 ...
2
votes
1answer
90 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
101 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
70 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
1answer
141 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
114 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 ...
16
votes
0answers
202 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
3
votes
3answers
77 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
74 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
30 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
56 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
52 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 ...
0
votes
0answers
18 views

transformation involving elliptic integrals

I have two expressions which I know are equivalent but I just can't see how to go from one to another. I'm sure it involves properties of elliptic integrals however I am not very familiar with the ...
3
votes
1answer
103 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
190 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
52 views

Limit evaluation with elliptic integrals

Prove the following involving elliptic integrals: $$ \lim_{u\to 0 } \dfrac{K(u)- E(u) } {1 - \sqrt {1-u}} = \frac{\pi}{2} $$
1
vote
1answer
53 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
65 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. ...
5
votes
0answers
54 views

About a sequence related with the complete elliptic integral of the second kind

When answering this related question I proved that if we define $B(\lambda)$ as: $$\begin{eqnarray*} ...
1
vote
4answers
102 views

How to find the area enclosed by the ellipse $b^2x^2 + a^2y^2=a^2b^2$?

I need to find the area enclosed by the ellipse $b^2x^2 + a^2y^2=a^2b^2$, and I know it involves taking the integral, but I'm not sure what function I should be taking the integral of or how to find ...
2
votes
1answer
43 views

Are these two elliptic integral evaluations identical?

I'm reading a paper on the Schwarz D minimal surface, and I'm wondering whether the authors have made a mistake. They evaluate the integral $$ \int_0^z \frac{2t\;\mathrm{d}t}{\sqrt{t^8-14t^2+1}}, $$ ...
0
votes
1answer
34 views

Can someone verify my derivation of a differential equation involving elliptic integrals, please?

I'm trying to determine the relationship between the major and minor radii ($a$ and $b$, respectively) of an ellipse of constant perimeter and variable eccentricity, and I've been thinking that ...
2
votes
0answers
182 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
1
vote
0answers
57 views

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 $[0,1].$ ...
1
vote
1answer
67 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 ...
1
vote
0answers
54 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 ...
2
votes
2answers
240 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 ...
2
votes
1answer
84 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 ...
3
votes
1answer
113 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 ...
4
votes
2answers
140 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) ...
0
votes
0answers
65 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 ...
2
votes
1answer
118 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 ...
0
votes
1answer
65 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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votes
0answers
59 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.
9
votes
1answer
163 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} ...
3
votes
0answers
57 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 ...
0
votes
1answer
32 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( ...
6
votes
1answer
106 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 ...
7
votes
1answer
201 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 ...
1
vote
0answers
41 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: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
17
votes
1answer
353 views
1
vote
0answers
73 views

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} ...
2
votes
1answer
75 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 ...