Questions tagged [elliptic-integrals]

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

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9 votes
3 answers
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Given algebraic $a$, find the closed form of $\int_0^a \dfrac{dx}{\sqrt{1-x^4}}$

Let $$A=\int_0^1 \dfrac{dx}{\sqrt{1-x^4}}.$$ Given an algebraic number $0\le a\le 1$, can we determine if there exists a rational number $b$ such that $$\int_0^a \dfrac{dx}{\sqrt{1-x^4}}=Ab?$$ If so, ...
Nomas2's user avatar
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6 votes
0 answers
91 views

Prove $\int_{0}^{1}\frac1k K(k)\ln\left[\frac{\left(1+k \right)^3}{1-k} \right]\text{d}k=\frac{\pi^3}{4}$

Is it possible to show $$ \int_{0}^{1}\frac{K(k)\ln\left[\tfrac{\left ( 1+k \right)^3}{1-k} \right] }{k} \text{d}k=\frac{\pi^3}{4}\;\;? $$ where $K(k)$ is the complete elliptic integral of the first ...
Setness Ramesory's user avatar
0 votes
0 answers
68 views

Solving the Elliptic Integral of the First Kind with Imaginary Moduli

The Complete Elliptic Integral of the First Kind is defined as the following: $$K(m)=\int_0^{\frac{\pi}{2}}\frac{dx}{\sqrt{1-m\sin^2(x)}}$$ Given this definition, I would like to solve the following ...
Oiler's user avatar
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1 vote
1 answer
67 views

Requesting clarification of Galois' last letter

I am reading Galois' last letter to Auguste Chevalier before the infamous duel. https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100 What is the notation H' for a group H supposed to represent ...
vallev's user avatar
  • 133
1 vote
0 answers
58 views

Aircraft Wing Analysis - Elliptical Pressure Distribution

I'm having great difficulty with generating the shear and bending moment diagrams for a wingspan. I am using the elliptical pressure distribution equation; however, this problem doesn't consider a ...
NauticalDave's user avatar
2 votes
1 answer
109 views

Evaluation or simplification of $\displaystyle{\int_0^{\frac{\pi}{4}}}\dfrac{dx}{\sqrt{A-\cos x-Bx}}$

The period for an inverted pendulum on which some external forces act is expressed in terms of its initial angle, $\Phi_o\in[0º,70º]$ (higher angles are unstable), as $$\mathcal{T}(\Phi_o)=2\sqrt 2\...
Joan S. Guillamet F.'s user avatar
0 votes
3 answers
112 views

Elliptic Integrals involving square roots of polynomial of fourth order

Does any one know the solution of the integral \begin{equation} \int_{a}^{u}\frac{x^{2}}{\sqrt{x(x-a)(x-b)(x-c) }}.dx \end{equation} where $u > a > b > 0 > c$ I believe it has one ...
pwm's user avatar
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7 votes
0 answers
404 views

Finding a closed form for the series $\sum_{n=0}^{\infty}\frac{\binom{2n}{n}^3}{64^n(n+1)^k}$ for $k=1,2,3,4$

Context: This question is related to Calculate $\sum_{n = 0}^\infty \frac{C_n^2}{16^n}$ and Is there a closed form for a give infinite sum?. We have also: $$\sum_{n=0}^{\infty}\frac{\binom{2n}{n}^3}{...
User's user avatar
  • 407
1 vote
1 answer
71 views

Trasformation of elliptic integral $\Pi(n;x|m)$ in function of $F(x|m)$ and $E(x|m)$

I everyone, I calculated these two formulas for $m<1$ and $z\in\mathbb{C}$: $$K(m)=F\left(\left.\frac{\pi}{2}-z\right|m\right)+\frac{1}{\sqrt{1-m}}\cdot F\left(z\left|\frac{m}{m-1}\right.\right)$$ $...
Math Attack's user avatar
  • 2,373
3 votes
1 answer
257 views

Conjectured closed forms for Eisenstein-like series

This question is related to: Eisenstein sum. Being $q=e^{\pi}$, we have also: \begin{align*} \sum_{n=1}^{\infty}\frac{n(q^{n}(-1)^{n}+1)}{q^{2n}+2(-1)^{n}q^{n}+1}=-\frac{1}{24}\tag{1}, \end{align*} \...
User's user avatar
  • 407
0 votes
0 answers
18 views

B-cycle integral with multi-valued function

Hy, I would like some help with my integral : $ \oint_B\frac{1}{z^2}\sqrt{(z-a)(z+\bar{a})(z-\frac{1}{a})(z+\frac{1}{\bar{a}})} $ Where B is the contour that connect the two branch cuts associated to ...
Yasser05batna's user avatar
2 votes
1 answer
73 views

Series related to $\,_{2p+1}F_{2p}\left(\left\{\frac{a}{2}\right\}_{2p-1},a,\frac12;a+\frac{1}{2},\left\{1+\frac{a}{2}\right\}_{2p-1};1\right)$

Similar things have been discussed here. As an example, let $p=2,a=1/3$ to have $$ {}_5F_4\left ( \frac16,\frac16,\frac16,\frac13,\frac12; \frac56,\frac76,\frac76,\frac76;1 \right )=\sum_{n=0}^{\infty}...
Setness Ramesory's user avatar
22 votes
2 answers
433 views

A closed form for a triple integral involving Heron's formula

Let $$S(x,y,z)=\frac14\sqrt{(x+y+z) (-x+y+z) (x-y+z) (x+y-z) }\tag1$$ (note that it's Heron's formula for the area of a triangle with sides of lengths $x,y,z$). I'm trying to evaluate the following ...
Vladimir Reshetnikov's user avatar
1 vote
0 answers
47 views

Splitting the Complete Elliptic Integral of the First Kind into real and imaginary functions

Letting $$K(m)=\int_0^\frac{\pi}{2}\frac{\mathrm du}{\sqrt{1-m\sin^2 u}}$$ be the Complete Integral of the First Kind, is there such an identity such that $K(z)=K(x+iy)=f(x)+ig(y)$?
Oiler's user avatar
  • 11
15 votes
3 answers
339 views

Convert integral $\int_0^{\frac{\sqrt3}2} {\frac{1}{{{x^{3/4}}{{\left( {1 + x} \right)}^{3/4}}}}dx}$ to elliptic integral

I found this integral from a friend of mine $$I = \int_0^{\frac{\sqrt3}2} {\frac{1}{{{x^{3/4}}{{\left( {1 + x} \right)}^{3/4}}}}dx} $$ Which its closed-form is :$$\frac{{2\sqrt 2 {\pi ^{3/2}}}}{{3{\...
OnTheWay's user avatar
  • 2,052
0 votes
0 answers
23 views

Lateral Surface Area for Elliptical Cone.

How to find the lateral surface of elliptical cone? Equation of lateral surface is Lateral surface equation but how to find the value of E in this equation? Here is an example. Example
Abu Bakar's user avatar
6 votes
2 answers
320 views

An integral related to Gamma value.

We have: $$\int_{0}^{\pi/2}\frac{\sin{x}\log{(\tan{(x/2))}+x}}{\sqrt{\sin{x}}(\sin{x}+1)}dx=\pi-\frac{\sqrt{2\pi}\Gamma{(1/4)}^{2}}{16}-\frac{\sqrt{2}\pi^{5/2}}{2\Gamma{(1/4)}^{2}}\tag{1}.$$ As other ...
User's user avatar
  • 407
0 votes
1 answer
29 views

Explicit solution for Harmonic Oscillators "solution in quadratures"

So I was reading about integrable systems and of course the harmonic oscillator in one dimension was mentioned as an example. After some simple calculations the author ended up with the following ...
Wurstcake's user avatar
  • 154
0 votes
0 answers
46 views

Series expansion of the Elliptic integral of second kind

I want to calculate the series expansion of the function $E(1-x)$, where $E$ is the complete elliptic integral of second kind defined as $$ E(x)=\int_0^{\pi/2} d\theta\, \sqrt{1-x^2\sin^2\theta} $$ ...
yglena's user avatar
  • 155
4 votes
1 answer
98 views

Finding a concise relation between $\operatorname{ns}\left(\frac{K(k)}{3},k\right)$ and $\operatorname{ns}\left(\frac{2K(k)}{3},k\right)$

Let $\operatorname{ns}\left(z,k\right)$ be one of the Jacobi's elliptic functions, and $K(k)$ the complete elliptic integral of the first kind. It's well-known that $\operatorname{ns}(mK(k),k)$ where $...
Setness Ramesory's user avatar
7 votes
2 answers
678 views

Integrals of elliptic integrals.

Being: $$R(k)=\left(-\frac{K(k)^2}{k}+K(k)E(k)\left(\frac{1}{k}-\frac{k}{2(k^2-1)}\right)\right)$$ We have these evaluations: $$\int_{0}^{1/\sqrt{2}}R(k)dk=\frac{3\Gamma{(1/4)}^4}{128\pi}-\frac{\pi^2}{...
User's user avatar
  • 407
1 vote
0 answers
45 views

Is there an algebraic function that satisfies the Abelian complex multiplication with even irrational multiplicator?

According to Abel's astonishing paper "RECHERCHES SUR LES FONCTIONS ELLIPTIQUES.[1]", we can get \begin{align*} \varphi_3(x,\kappa_3)&=\frac{x\cdot\left(\sqrt{3}-\,\kappa_3x^{2}\right)}{...
Eufisky's user avatar
  • 3,063
2 votes
1 answer
194 views

Eisenstein sum.

I have a proof of: $$S=\sum_{n=1}^{\infty}\frac{n(-1)^{n+1}}{(-1)^n+e^{\pi n}}=\frac{1}{24}.$$ That is related to Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}...
User's user avatar
  • 407
0 votes
0 answers
61 views

Evaluating Function of Incomplete Elliptic Integrals

I am trying to write Mathematica code that evaluates the following function: $$ f(\kappa_{yx}, \kappa_{zx}) = 1 + 3 \kappa_{yx} \kappa_{zx} \frac{E(\varphi \backslash \alpha) - F(\varphi \backslash \...
steveaw123801's user avatar
1 vote
1 answer
55 views

Express $\int^d_c \frac{\mathrm{d}u}{\sqrt{b^2\sinh^2 u -a}}du$ in terms of elliptic integral.

I want to express $$\int^d_c \frac{\mathrm{d}u}{\sqrt{b^2\sinh^2 u -a}}du$$ (where $a,b,c,d$ are real constants) in terms of elliptic functions. I have tried to make the naive change of variable $b^2\...
Grimolatto's user avatar
1 vote
1 answer
53 views

Questions about reduction of elliptic integrals and general calculus

I am working on a numerical problem and trying to simplify elliptic integrals to reduce computational costs. I found a previous question that is similar to my case, but I don't understand part of it. ...
noon's user avatar
  • 11
-1 votes
1 answer
50 views

Simplifying Elliptic Integrals 2 [closed]

According to wolfram, the following equation holds. However, I do not understand the derivation process. Could you please tell me how to derive it? $$\int_0^1 \frac{u^4}{\sqrt{(1 - u^2) (1 - k u^2)}} ...
noon's user avatar
  • 11
0 votes
1 answer
53 views

Simplifying Elliptic Integrals

According to wolfram alpha, the following equation holds. $$\int_0^1 \frac{ku^2}{\sqrt{(1 - u^2) (1 - k u^2)}}=K(k)-E(k)$$ where $K$ and $E$ are complete elliptic integrals of the first and second ...
noon's user avatar
  • 11
2 votes
0 answers
64 views

Integrating $\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$.

$$\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$$ I know we have $E,K,$ and $\Pi$ to integrate quartics under the radical, but I wonder what functions quintics yield? Are there any known?
Alexander Conrad's user avatar
7 votes
2 answers
311 views

Evaluate $I(a,b)=\int_{0}^{1}k^a(1-k^2)^bK(k)\text{d}k$

With the interests of $$ I(a,b)=\int_{0}^{1}k^a(1-k^2)^bK(k)\text{d}k, $$ where $K(k)$ represents the complete elliptic integral with modulus $k$ and $K^\prime(k)$ its complementary, many $I(a,b)$ for ...
Setness Ramesory's user avatar
12 votes
1 answer
920 views

Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$

The paper mentioned a proposition: $$ \int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }. $$ Its equivalent is $$ \int_{0}^{\infty}\vartheta_2(...
Setness Ramesory's user avatar
6 votes
1 answer
113 views

Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$

I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$): \begin{aligned} \int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4 \text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
Setness Ramesory's user avatar
3 votes
1 answer
54 views

Expansion of elliptic integral close to a critical point

I am looking at the Integral $$\int_{a_2}^{a_1} R(x) \sqrt{a_1-x}\sqrt{x-a_2}\sqrt{x-a_3} dx$$ with $$a_1>a_2>a_3$$ and $R(x)$ is a rational function of $x$ with a pole of second order in $x=c$ ...
Phil Ler's user avatar
1 vote
0 answers
47 views

How to show that $\int_0^1 \frac{1}{\sqrt{x(1-x)(x+c) }}=\frac{2K(-\frac{1}{c})}{\sqrt{c}}$?

From WolframAlpha or Mathematica $$\int_0^1 \frac{1}{\sqrt{x(1-x)(x+c) }}=\frac{2K(-\frac{1}{c})}{\sqrt{c}}$$ for $c>0$ where $$K(x)$$ was the complete elliptic integral of the first kind. However, ...
ShoutOutAndCalculate's user avatar
7 votes
0 answers
267 views

Evaluate $\int_{0}^{1} \frac{K(k)E(k)^2-\frac{\pi^3}{8} }{k} \text{d}k$ and $\int_{0}^{1} \frac{E(k)^3-\frac{\pi^3}{8} }{k} \text{d}k$

Let $K(k),E(k)$ be the complete elliptic integral of the first kind and second kind respectively, where $k$ is the elliptic modulus. Consider four integrals, $$\begin{aligned} &I_1=\int_{0}^{1} \...
Setness Ramesory's user avatar
1 vote
1 answer
100 views

How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2 -4 \cos^2\theta}}\ d\theta$ [closed]

Spent several days already trying to figure out how to convert the given integral to this: $$\frac{8}{\sqrt{(\eta-1)^3(\eta+3)}}K\left(\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}}\right)$$ where $$ K\left(...
Sombercy's user avatar
11 votes
2 answers
294 views

Integrals of $\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$

Consider a type of integrals $$ \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k $$ where $K=K(k),K^\prime=K(\sqrt{1-k^2})$ are complete elliptic integrals, and $k$ is an elliptic ...
Setness Ramesory's user avatar
1 vote
1 answer
139 views

Are there any simpler transformations for the elliptic integral of the second kind $\text E(z,m)$ than these?

$\def\F\{\operatorname F}\def\E{\operatorname E}$ Elliptic F$(x,m)$ appears in elliptic E$(x,m) =\int_0^x\sqrt{1-m\sin^2(x)}dx$ parameter $m$ transformations. Luckily the DLMF has reciprocal/imaginary ...
Тyma Gaidash٠'s user avatar
3 votes
1 answer
120 views

How to cancel this Jacobi amplitude, elliptic integral, and incomplete beta function composition?

Problem and Context: $\def\K{\operatorname K}\def\F{\operatorname F} \def\sn{\operatorname{sn}} \def\B{\operatorname B} \def\I{\operatorname I} \def\E{\operatorname E}\def\am{\operatorname{am}}$ An ...
Тyma Gaidash٠'s user avatar
2 votes
1 answer
85 views

Evaluating elliptic integral of 2nd kind with added secant

I'm trying to evaluate the integral $$\int_{0}^{\phi} \frac{1}{2}\sec\left(\frac{x}{2}\right)\sqrt{1+\sin^2 (x)}\, dx$$ For $\phi$ between 0 and $\pi$. I know how to evaluate it without the secant, ...
Da Monster's user avatar
1 vote
2 answers
65 views

Can you explain why $du/\sqrt{c-2u} = d\varepsilon$ using hyperbolic substitution integral becomes $A\operatorname{sech}^2(t)$?

Here I have a 3rd order ODE wave equation $$ -cu' + 6uu' + u''' = 0 $$ where $u(\varepsilon) = u$ and $\varepsilon = x - ct$; a wave (assume $\varepsilon$ is one variable; hence $u' = du/d\varepsilon$)...
Jeremy Zelic's user avatar
6 votes
4 answers
207 views

A theta function identity involving $\vartheta_2(q^3),\vartheta_3(q^3)$

How can we verify the theta function identity? $$ \left ( \vartheta_2(q)^2+3\vartheta_2(q^3) ^2\right )\left ( \vartheta_3(q)^2+3\vartheta_3(q^3)^2 \right )=4\vartheta_2(q)^2\vartheta_3(q)^2. $$ Where ...
Setness Ramesory's user avatar
7 votes
2 answers
381 views

Evaluate $\int_{0}^{\pi/2} \ln\left[ \tan\left ( \frac{\theta}{2}\right) \right ]^2 K\left ( \sin\theta \right )\text{d}\theta$

Let us define $K(x)$ as complete elliptic integral of the first kind, where $x$ is elliptic modulus. A possible closed-form is ($G$ denotes Catalan's constant.) $$ \int_{0}^{\pi/2} \ln\left[ \tan\left ...
Setness Ramesory's user avatar
0 votes
0 answers
49 views

Closed form for recursive square super-roots of two numbers?

Take two real numbers $x$ and $y$ greater than one. Set $x_0=x$ and $y_0=y$, and iterate the following: $$x_{n+1}=e^{W(y_n\text{ln}(x_n))}=\text{ssrt}(x_n^{y_n}),y_{n+1}=e^{W(x_n\text{ln}(y_n))}=\text{...
ln 23.14's user avatar
4 votes
1 answer
126 views

Are these sources wrong about Jacobi elliptic functions, and can we fix the flaw?

I first noticed that three Youtube videos [1] [2] [3] give the same definitions of the Jacobi elliptic functions sn, cn, dn (see below), which are different from Wikipedia's. Then I noticed the links ...
WillG's user avatar
  • 6,422
8 votes
5 answers
566 views

Evaluation of $\int_0^1\frac{\log x\,dx}{\sqrt{x(1-x)(1-cx)}}$

Assume $c$ is a small real number. QUESTION. What is the value of this integral in terms of the complete elliptic function $K(k)$? $$\int_0^1\frac{\log x}{\sqrt{x(1-x)(1-cx)}}\,dx.$$ I got as far as ...
T. Amdeberhan's user avatar
2 votes
2 answers
141 views

Express $\sum_{n\in\mathbb{Z}} \left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right )$

How can we find a expression for the following sum $$ S=\sum_{n\in\mathbb{Z}} \left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right ) $$ where $q=e^{-\pi{K^\prime(k)}/{K(k)}}$ and $K(x)=\int_{0}^{1} \frac{1}{\sqrt{...
Setness Ramesory's user avatar
2 votes
1 answer
79 views

Can order of summation and integral be interchanged in : $\int_{-1}^1 ( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime))\text{d}\xi^\prime$?

I am wanting to know if there is a proof that the order of summation and integration can be interchanged in $$\int_{-1}^1 \left( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime)\right)\text{d}\...
Joseph Robert Jepson's user avatar
0 votes
0 answers
21 views

Arc length of a graph of an elliptic function

Exercise 2 on p. 190 of Greenhill's ''Applications of Elliptic Functions'' asks me to rectify, by means of elliptic arcs, $y=\sin x$ and $y=\operatorname{sn}(x,k).$ For $y=\sin x$, I get $$\int \...
Vestoo's user avatar
  • 407
2 votes
1 answer
297 views

Derivation of an integral containing the complete elliptic integral of the first kind

This is a repost of mathoverflow to draw broader attentions. https://mathoverflow.net/questions/439770/derivation-of-an-integral-containing-the-complete-elliptic-integral-of-the-first I found the ...
r-nishi's user avatar
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