Questions tagged [elliptic-integrals]
Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.
607
questions
9
votes
3
answers
149
views
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, ...
- 91
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 ...
- 4,277
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 ...
- 11
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 ...
- 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 ...
- 11
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
$$\mathcal{T}(\Phi_o)=2\sqrt 2\...
0
votes
3
answers
112
views
Elliptic Integrals involving square roots of polynomial of fourth order
Does any one know the solution of the integral
\begin{equation}
\int_{a}^{u}\frac{x^{2}}{\sqrt{x(x-a)(x-b)(x-c) }}.dx \end{equation}
where $u > a > b > 0 > c$
I believe it has one ...
- 11
7
votes
0
answers
404
views
Finding a closed form for the series $\sum_{n=0}^{\infty}\frac{\binom{2n}{n}^3}{64^n(n+1)^k}$ for $k=1,2,3,4$
Context:
This question is related to Calculate $\sum_{n = 0}^\infty \frac{C_n^2}{16^n}$ and Is there a closed form for a give infinite sum?.
We have also:
$$\sum_{n=0}^{\infty}\frac{\binom{2n}{n}^3}{...
- 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 $z\in\mathbb{C}$:
$$K(m)=F\left(\left.\frac{\pi}{2}-z\right|m\right)+\frac{1}{\sqrt{1-m}}\cdot F\left(z\left|\frac{m}{m-1}\right.\right)$$
$...
- 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*}
\...
- 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 ...
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
$$
{}_5F_4\left ( \frac16,\frac16,\frac16,\frac13,\frac12;
\frac56,\frac76,\frac76,\frac76;1 \right )=\sum_{n=0}^{\infty}...
- 4,277
22
votes
2
answers
433
views
A closed form for a triple integral involving Heron's formula
Let
$$S(x,y,z)=\frac14\sqrt{(x+y+z) (-x+y+z) (x-y+z) (x+y-z) }\tag1$$
(note that it's Heron's formula for the area of a triangle with sides of lengths $x,y,z$).
I'm trying to evaluate the following ...
- 46.8k
1
vote
0
answers
47
views
Splitting the Complete Elliptic Integral of the First Kind into real and imaginary functions
Letting
$$K(m)=\int_0^\frac{\pi}{2}\frac{\mathrm du}{\sqrt{1-m\sin^2 u}}$$
be the Complete Integral of the First Kind, is there such an identity such that
$K(z)=K(x+iy)=f(x)+ig(y)$?
- 11
15
votes
3
answers
339
views
Convert integral $\int_0^{\frac{\sqrt3}2} {\frac{1}{{{x^{3/4}}{{\left( {1 + x} \right)}^{3/4}}}}dx}$ to elliptic integral
I found this integral from a friend of mine
$$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{\...
- 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
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 ...
- 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 ...
- 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
$$
E(x)=\int_0^{\pi/2} d\theta\, \sqrt{1-x^2\sin^2\theta}
$$
...
- 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 $...
- 4,277
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:
$$\int_{0}^{1/\sqrt{2}}R(k)dk=\frac{3\Gamma{(1/4)}^4}{128\pi}-\frac{\pi^2}{...
- 407
1
vote
0
answers
45
views
Is there an algebraic function that satisfies the Abelian complex multiplication with even irrational multiplicator?
According to Abel's astonishing paper "RECHERCHES SUR LES FONCTIONS ELLIPTIQUES.[1]", we can get
\begin{align*}
\varphi_3(x,\kappa_3)&=\frac{x\cdot\left(\sqrt{3}-\,\kappa_3x^{2}\right)}{...
- 3,063
2
votes
1
answer
194
views
Eisenstein sum.
I have a proof of:
$$S=\sum_{n=1}^{\infty}\frac{n(-1)^{n+1}}{(-1)^n+e^{\pi n}}=\frac{1}{24}.$$
That is related to Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}...
- 407
0
votes
0
answers
61
views
Evaluating Function of Incomplete Elliptic Integrals
I am trying to write Mathematica code that evaluates the following function:
$$
f(\kappa_{yx}, \kappa_{zx}) = 1 + 3 \kappa_{yx} \kappa_{zx} \frac{E(\varphi \backslash \alpha) - F(\varphi \backslash \...
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\...
- 159
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.
...
- 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?
$$\int_0^1 \frac{u^4}{\sqrt{(1 - u^2) (1 - k u^2)}} ...
- 11
0
votes
1
answer
53
views
Simplifying Elliptic Integrals
According to wolfram alpha, the following equation holds.
$$\int_0^1 \frac{ku^2}{\sqrt{(1 - u^2) (1 - k u^2)}}=K(k)-E(k)$$
where $K$ and $E$ are complete elliptic integrals of the first and second ...
- 11
2
votes
0
answers
64
views
Integrating $\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$.
$$\int \frac{dx}{\sqrt{x^5+ax^3+bx^2+cx+d}}$$
I know we have $E,K,$ and $\Pi$ to integrate quartics under the radical, but I wonder what functions quintics yield? Are there any known?
- 1,495
7
votes
2
answers
311
views
Evaluate $I(a,b)=\int_{0}^{1}k^a(1-k^2)^bK(k)\text{d}k$
With the interests of
$$
I(a,b)=\int_{0}^{1}k^a(1-k^2)^bK(k)\text{d}k,
$$
where $K(k)$ represents the complete elliptic integral with modulus $k$ and $K^\prime(k)$ its complementary,
many $I(a,b)$ for ...
- 4,277
12
votes
1
answer
920
views
Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$
The paper mentioned a proposition:
$$
\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(...
- 4,277
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\...
- 4,277
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 $x=c$ ...
- 31
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, ...
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,
$$\begin{aligned}
&I_1=\int_{0}^{1} \...
- 4,277
1
vote
1
answer
100
views
How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2 -4 \cos^2\theta}}\ d\theta$ [closed]
Spent several days already trying to figure out how to convert the given integral to this: $$\frac{8}{\sqrt{(\eta-1)^3(\eta+3)}}K\left(\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}}\right)$$
where
$$
K\left(...
- 11
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 ...
- 4,277
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 ...
- 10.1k
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 ...
- 10.1k
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, ...
- 23
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 $u' = du/d\varepsilon$)...
- 21
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 ...
- 4,277
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.)
$$
\int_{0}^{\pi/2}
\ln\left[ \tan\left ...
- 4,277
0
votes
0
answers
49
views
Closed form for recursive square super-roots of two numbers?
Take two real numbers $x$ and $y$ greater than one. Set $x_0=x$ and $y_0=y$, and iterate the following:
$$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{...
- 71
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 ...
- 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 ...
- 303
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{...
- 4,277
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 $$\int_{-1}^1 \left( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime)\right)\text{d}\...
0
votes
0
answers
21
views
Arc length of a graph of an elliptic function
Exercise 2 on p. 190 of Greenhill's ''Applications of Elliptic Functions'' asks me to
rectify, by means of elliptic arcs, $y=\sin x$ and $y=\operatorname{sn}(x,k).$
For $y=\sin x$, I get
$$\int \...
- 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 ...
- 23