Skip to main content

Questions tagged [elliptic-integrals]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
28 views

Analogues of elliptic functions for generalisations of elliptic integrals

I'm looking at the following integral $t = \int_{x_0}^{x(t)} \frac{dx'} {\sqrt{A+Bx'+Cx'^2+Dx'^3+Ex'^4+Fx'^6}}$ and want to know what the inverse function $x(t)$ is, knowing $x(0)=x_0$ and $A=-...
Baenazril's user avatar
0 votes
0 answers
12 views

How to transform elliptic integrals with conjugate complex variables into a linear sum of elliptic integrals with real variables and imaginary units?

How to transform elliptic integrals with conjugate complex variables into a linear sum of elliptic integrals with real variables and imaginary units? It seems wrong to use the addition formula ...
D.Matthew's user avatar
  • 907
0 votes
1 answer
40 views

Elliptic integral of complex argument with value greater than unity

I need to compute the elliptic integral with a complex argument. I know that there exists a formula for the incomplete elliptic integral of the first kind for complex arguments, as follows from P.F....
won5830's user avatar
  • 103
3 votes
0 answers
95 views

Proving the relation between elliptic integral squared and ${}_3F_2$

I want to prove that $$\frac{4}{\pi^2}K^2=\frac{1}{1+k^2}\sum_{n=0}^{\infty} \frac{(4n)!}{n!^4}\left(\frac{k(k^2-1)}{4(k^2+1)^2}\right)^{2n}$$ See notation reference here Different approach to the sum ...
Dqrksun's user avatar
  • 574
1 vote
0 answers
63 views

How to express the definite integral of this hyperelliptic integral as the definite integral of the elliptic integral

How to express the definite integral of this hyperelliptic integral as the definite integral of the elliptic integral? Using the substitution formula \begin{cases} p=e^{-\frac{2 i \pi }{5}} t^2+\frac{...
Eufisky's user avatar
  • 3,267
0 votes
0 answers
43 views

On-Axis Magnetic Field of a Finite Continuous Solenoid

I am attempting to verify the equation for the magnetic field on axis of a finite continuous solenoid posted to this wikipedia page. The equation is $$ B_z = \frac{\mu_0 NI}{2} \left( \frac{\frac{l}{2}...
Bunji's user avatar
  • 113
3 votes
0 answers
49 views

Evaluate $\operatorname{nd}\left ( \frac{2K_6^\prime}{3},k_6^\prime \right ) \operatorname{sd} \left ( \frac{2K_6^\prime}{3} ,k_6^\prime \right )$

Could we get a simple expression for $$ \operatorname{nd}\left ( \frac{2K_6^\prime}{3},k_6^\prime \right ) \operatorname{sd} \left ( \frac{2K_6^\prime}{3} ,k_6^\prime \right ), $$ $$ k_6^\prime=\sqrt{...
Setness Ramesory's user avatar
3 votes
2 answers
146 views

Show that $\int_0^1K^2(k)dk=\frac12\int_0^1K'^2(k)dk$

By switching integrals in double integral, I showed that $$\int_0^1K(k)dk=\int_0^1K'(k)dk=2G$$ where $K(k)$ is complete elliptic integral of the first kind and $K'(k)=K(\sqrt{1-k^2})$ is its ...
Bob Dobbs's user avatar
  • 11.9k
1 vote
1 answer
48 views

Elliptic integral singular expansion

The question. Consider the Elliptic Integral $$ F(x;k)=\int_0^x \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}}.\tag{1}\label{1} $$ I am interested in the singular series expansion of $F(1;k)$ about $k=1$. I was ...
Gateau au fromage's user avatar
8 votes
1 answer
208 views

Generating function for the products of pairs of Narayana numbers

The Narayana numbers OEIS sequence A001263 are given by: $$\operatorname{N}(n, k) = \frac{1}{n} {n \choose k} {n \choose k-1},$$ and have the generating function: $$G(z,t) = \sum_{n=1}^\infty \sum_{k=...
maxwelldecoherence's user avatar
2 votes
0 answers
38 views

Generalization of elliptic theta function?

We have the elliptic theta function of third kind $$ \vartheta_3(z)=\sum_{n=-\infty}^{\infty}z^{n^2} $$ Is there a generalization of this function to other exponents? I am looking for $$ \sum_{n=-\...
Roman's user avatar
  • 707
3 votes
1 answer
136 views

An "almost" geodesic dome

A regular $ n$-gon is inscribed in the unit circle centered in $0$. We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
user967210's user avatar
1 vote
2 answers
99 views

A formula for the $n^\text{th}$ power of elliptic-like integrands.

$$ \mbox{Define}\ \operatorname{F}\left(k,t\right) := \cos^{2}\left(t\right) + k^{2}\sin^{2}\left(t\right)\ \mbox{and}\ \operatorname{I}\left(n,k\right) := \frac{1}{\pi}\int_{0}^{\pi/2}\left[\...
Awe Kumar Jha's user avatar
8 votes
1 answer
189 views

Different approach to the sum $\sum_{n=0}^{\infty}\frac{(4n)!}{n!^4} \frac{1}{4^{4n}}$

I am trying to study the function $$ V(x)=\sum_{n=0}^{\infty}\frac{(4n)!}{n!^4}\frac{x^{2n}}{16^{2n}} $$ Using the Beta function, one can prove that $$ V(x)=\frac{4}{\pi^2}\int_0^1 \frac{K(xt^2)}{...
Dqrksun's user avatar
  • 574
6 votes
2 answers
192 views

How to calculate $\int_{0}^{2\pi}\frac{\sin^{2}k}{(1+a\cos k)\sqrt{1+b\cos k}}dk$ (elliptic integral)?

My goal is to calculate $$\int_{0}^{2\pi}\frac{\sin^{2}k}{(1+a^2+2a\cos k)\sqrt{1+b^2+2b\cos k}}dk,$$ where $a\neq b$ and $b\neq 1$. But we may simplify it into $\int_{0}^{2\pi}\frac{\sin^{2}k}{(1+a\...
ZJX's user avatar
  • 337
12 votes
2 answers
282 views

Bauer's series for $\frac{1}{\pi}$

Recently, someone asked a question involving the expression $$ \sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$ At first glance, I knew that the expression was the value of ...
seoneo's user avatar
  • 1,891
3 votes
3 answers
119 views

Derive the use of the Jacobi Amplitude function with the nonlinear pendulum diffequation

I've been playing around with the pendulum differential equation ${\theta}''+\frac{g}{L}\sin{\theta}=0$, and have found many general solutions using the Jacobi Amplitude function $\text{am}(u,m)$ ...
Kareem Shamma's user avatar
2 votes
1 answer
38 views

An integral equality about an incomplete elliptic integral of the first kind

If $|x|<1$ and $x\sin\alpha=\sin(2\beta-\alpha), 0\leq \alpha\leq \pi, 0\leq \beta\leq \pi/2,$ then $$(1+x)\int_{0}^{\alpha}\frac{d\phi}{\sqrt{1-x^2\sin^2\phi}}=\int_{0}^{\beta}\frac{d\phi}{\sqrt{1-...
Nick's user avatar
  • 521
4 votes
2 answers
97 views

Convert an equation to its elliptic form

I am failing to convert $L = \frac{\sqrt{2}}{2}\int_{0}^{\phi_0} \frac{\sin(\phi)}{\sqrt{\sin \phi_0 - \sin \phi}} d\phi$ into $L=\int_{\theta_1}^{\pi/2}\frac{2k^2\sin^2(\theta)-1 }{\sqrt{1-k^2\sin^2 \...
victor's user avatar
  • 155
2 votes
0 answers
238 views

Showing $\int_{0}^{1}\frac{E(\tfrac{x}{\sqrt{x^2+8}})}{\sqrt{8-7x^2-x^4}}dx=\frac{1}{3}K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})$

Context $\begin{align} K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1} \end{align}$ and $\begin{align} E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2t}dt\tag{2} \end{align}$ the complete elliptic ...
User's user avatar
  • 323
2 votes
2 answers
77 views

What is the correct argument definition for the complete elliptic integral?

For the complete elliptic integral of the first kind, we find on Wikipedia: $$ K(k) = \int_\limits{0}^{\pi/2} \hspace{-1ex} \frac{d\theta} {\sqrt{1 - k^2 \sin^2 \theta}} $$ But in Abramowitz and ...
Jos Bergervoet's user avatar
0 votes
2 answers
59 views

Finding an approximation for the complete elliptic integral of the first kind $K(k)$

Finding an approximation for the complete elliptic integral of the first kind $K(k)$ In this question the author find the following approximation to the integral: $$ I(k) := \int\limits_{0}^{\frac{\pi}...
Joako's user avatar
  • 1,586
4 votes
0 answers
85 views

Differential 1-forms over an (hyper)elliptic curve

Given am elliptic curve $E: y^2=x^3+ax+b$, a nice basis for $\Omega^1$ (for the holomorphic 1-forms) might be $\{\frac{dx}{y}\}$. Given a hyperelliptic curve $H: y^2=P(x)$ where $P(x)$ is of degree $...
Or Shahar's user avatar
  • 1,804
10 votes
1 answer
442 views

How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$

The formula $$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$ (in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...
Nomas2's user avatar
  • 667
3 votes
2 answers
200 views

How to reduce $\int_0^{\pi/2}\frac{1-\sin x}{\cos^2x}\sqrt{\tan x}\,dx$ to complete elliptic integral?

I came across another old post concerning a definite integral whose closed form can be expressed with a complete elliptic integral: $$I = \int_0^\infty \left(\sqrt{1+x^4} - x^2\right) \, dx = \frac1{6\...
user170231's user avatar
  • 20.6k
0 votes
1 answer
46 views

Estimating the parameters of an ellipse (part 3)

This post is a follow up of this and this previous ones. I've found an explanation for the following formulas \begin{equation} \hat{\ell}_1 \triangleq 2\sqrt{\hat{\Lambda}_{11}} \qquad \hat{\ell}_2 \...
matteogost's user avatar
1 vote
2 answers
41 views

Asymptotics of parametric integral using elliptic integrals

I am reading a physics paper, namely "Note on hydrodynamics" by Sir Charles Darwin (I know what you may think, it's not that Darwin) and a relevant quantity (a drift) can be expressed, as a ...
tommy1996q's user avatar
  • 3,366
1 vote
0 answers
50 views

The behavior of a kind of elliptical integral while one of its parameters turns to infinity.

This integral is developed in the study to 3-dimensional axisymmetric Biot-Savart operator, and could be written as below: $$K_{3d}(a,b,c,l):=\int_{0}^{2\pi}\frac{l\sqrt{(a+l)^2+(b+l)^2-2(a+l)(b+l)\...
Orangeeee's user avatar
  • 155
5 votes
2 answers
105 views

Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$

Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
bkocsis's user avatar
  • 1,258
2 votes
3 answers
140 views

Can we reduce $\int_0^{\pi/2}\frac{\sqrt{\sin x}}{1+\cos x}\,dx$ to complete elliptic integrals?

This definite integral has an equivalent closed form in terms of complete elliptic integrals, $$\begin{align*} I &= \int_0^\tfrac\pi2 \frac{\sqrt{\sin x}}{1+\cos x} \, dx \\ & = 2 - \sqrt{\...
user170231's user avatar
  • 20.6k
2 votes
4 answers
200 views

Show that $\int_0^{\frac\pi 2}\frac{\theta-\cos\theta\sin\theta}{2\sin^3\theta}d\theta=\frac{2C+1}4$

While trying to evaluate $\int_0^1 k^2K(k)dk$ related to elliptic integral of the first kind, by integral switching method, I reached the trigonometric integral $$\int_0^{\frac\pi 2}\frac{\theta-\cos\...
Bob Dobbs's user avatar
  • 11.9k
9 votes
1 answer
711 views

On Ramanujan's fastest series.

Context With some effort we can show that Ramanujan's fastest series implies: \begin{align} \frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1} \end{align} ...
User's user avatar
  • 323
1 vote
0 answers
69 views

Indefinite integral of elliptic integrals

Derivative of complete elliptic integrals, $E(k)$, $K(k)$, etc., are known. But, I don't know about their integrals. I tried to evaluate the indefinite integral $$\int_0^k K(k)dk\tag1$$ and ended up ...
Bob Dobbs's user avatar
  • 11.9k
4 votes
0 answers
75 views

Trigonometric Substitutions related to Elliptic Integrals

Recently while studying Elliptic Integrals and related topics I have come across various Interesting Trigonometric Substitutions, examples given below 1. $$\int_0^{\pi/2}\frac{1}{\sqrt{1-x^2(\sin t)^4}...
Miracle Invoker's user avatar
1 vote
1 answer
236 views

An integral related to $\Gamma(\tfrac14)^2$

I converted $\Gamma(\tfrac14)^2=16\Gamma(\tfrac54)^2$ to a double integral by definition of Gamma function. By using polar coordinates I eliminated one integral and found $12\sqrt{\pi}$ times the ...
Bob Dobbs's user avatar
  • 11.9k
3 votes
1 answer
253 views

Integral $\int_0^{1/\sqrt{2}}\frac{K\left[\sqrt{1-k^2}\right]}{\sqrt{1-2k^2}\sqrt{1-k^2}}dk$

$$\int_0^{1/\sqrt{2}}\frac{K\left[\sqrt{1-k^2}\right]}{\sqrt{1-2k^2}\sqrt{1-k^2}}dk=\frac{\Gamma^2(1/8)\Gamma^2(3/8)}{32\pi}$$ With $K$ as the Complete Elliptic Integral of the First Kind, $$K:=K(k)=\...
Miracle Invoker's user avatar
2 votes
1 answer
75 views

Is $\int_0^{\frac\pi 2}\frac{dx}{1-k^2\sin^2x}$ an elliptic integral?

Let $\Delta=\sqrt{1-k^2\sin^2x}$, $E(k)=\int_0^{\frac\pi 2}\Delta dx$ and $K(k)=\int\frac{dx}{\Delta}.$ Start wearing purple, gives a nice answer for the interesting identity $\int_0^{\frac\pi 2}\frac{...
Bob Dobbs's user avatar
  • 11.9k
1 vote
1 answer
38 views

Transformation of an integral of the form $\int R(x,\sqrt{f(x)})dx$ to a classical Jacobi elliptic integral

In a paper I have the following statement: The integral $$\phi_{\infty}= \frac{1}{\sqrt{2M}}\int_{0}^{\frac{1}{P}}\frac{dt}{\sqrt{G(t)}}$$ with $$G(t)= t^3 - \frac{1}{2M}t^2 + \frac{P-2M}{2MP^3}$$ can ...
Bufo Viridis's user avatar
1 vote
1 answer
65 views

How we get the jacobi elliptic functions [closed]

I am encountering an issue with Jacobi elliptic functions. Specifically, I am facing a challenge with the following integral: $$\pm\int_{-\sqrt{\frac{2\mu}{\eta}}}^{u}\frac{\mathrm ds}{s\sqrt{\left(s+\...
Espoir's user avatar
  • 15
1 vote
1 answer
72 views

The holomorphic differential of an elliptic curve as a Riemann surface

I am reading a Teichmueller theory book and trying to understand elliptic curves as examples of Riemann surfaces. Consider the elliptic curve \[X = \{[z : w : y] \in \mathbb C \mathrm P^2 \mid w^2y = ...
Chaitanya Tappu's user avatar
4 votes
0 answers
124 views

Is it possible to evaluate this integral? If not, is it possible to determine whether the result is an elliptic function or not?

I am trying to evaluate the integral $$F(x,y) = \int_0^1 du_1\, \int_0^{1-u_1} du_2\, \frac{\log f(x,y|u_1,u_2)}{f(x,y|u_1,u_2)}\,, \tag{1}$$ with $$f(x,y|u_1,u_2) := u_1(1-u_1)+y\, u_2(1-u_2) + (x-y-...
Pxx's user avatar
  • 697
6 votes
2 answers
176 views

Show that $\int_0^{\frac{\pi}2}K(\sin t)dt=\frac{\Gamma(\tfrac14)^4}{16\pi}$

By using the Maclaurin series $K(k)=\frac{\pi}2\sum_{n=0}^\infty c_n ^2 k^{2n}$ where $c_{n}={2n\choose n}2^{-2n}$ we have $$\int_0^{\frac{\pi}2}K(\sin t)dt\\ =\frac{\pi}2\sum_{n=0}^\infty c_n ^2\...
Bob Dobbs's user avatar
  • 11.9k
3 votes
0 answers
184 views

Direct evaluation challenge of a Dedekind $\eta$ elliptic integral

I believe I have a challenge for the integration community. Show directly: $$2\pi\int_{0}^{\infty}\frac{\eta(4ix)^6\eta(20ix)^6}{\eta(2ix)^2\eta(8ix)^2\eta(10ix)^2\eta(40ix)^2}\,\mathrm{d}x=\frac{1}{\...
KStar's user avatar
  • 5,337
2 votes
1 answer
136 views

Ramanujan-Type Double Sum Infinite Series for $1/\pi$

$$\sum_{n,m=0}^{\infty}\binom{2n}{n}\binom{2m}{m}\binom{2n+2m}{n+m}^3\left(\frac{1+6n+6m}{2^{10n+10m}}\right)=\frac{4}{\pi}$$ These are Ramanujan-Type Series for $1/\pi$ but based on a Double Sum ...
Miracle Invoker's user avatar
17 votes
0 answers
742 views

Other approaches to $\int_{0}^{1} \frac{K\left ( x \right ) }{\sqrt{3-x} } \text{d}x$

Let $K(x) = \int_0^{\pi/2}\frac{1}{\sqrt{1-x^2 \sin^2 \theta}}d\theta$ be the complete elliptic integral of first kind. It could be shown that $$ \int_{0}^{1} \frac{K\left ( x \right ) }{\sqrt{3-x} } \...
Setness Ramesory's user avatar
1 vote
1 answer
105 views

$\int_0^1E(k)dk$ without switching integrals

Let $E=E(k), K=K(k)$ be the complete elliptic integrals of the second and the first kind respectively. It is well-known that $\frac12\int_0^1K\,dk=G$ is the Catalan constant. We can also find, by ...
Bob Dobbs's user avatar
  • 11.9k
2 votes
0 answers
74 views

Strange algebra among elliptic moments $k^pK^qK^\prime{}^r$

Let us define the complete elliptic integral of the first kind as follows: $$ K(k)=\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-k^2t^2} } \text{d}t, $$ where we restrict the modulus $k$ to $|k|<1$ ...
Setness Ramesory's user avatar
3 votes
1 answer
156 views

Series $\sum_{n=0}^{\infty}(-1)^n\binom{2n}{n}^5\left(\frac{1+4n}{2^{10n}}\right)=\frac{\Gamma^4(1/4)}{2\pi^4}$

$$\sum_{n=0}^{\infty}(-1)^n\binom{2n}{n}^5\left(\frac{1+4n}{2^{10n}}\right)=\frac{\Gamma^4(1/4)}{2\pi^4}$$ Is there any way to prove this? I don't even know where to start with this one. The following ...
Miracle Invoker's user avatar
4 votes
2 answers
215 views

Integrate $\int_0^1\left(\int_0^\pi \frac{u}{\sqrt{1+u^2-2u \cos\phi}} d\phi\right)du$

I was trying to solve this integral while solving an electromagnetics problem in physics. $$\int_0^1\left(\int_0^\pi \frac{u}{\sqrt{1+u^2-2u \cos\phi}} d\phi\right)du$$ My approach My idea was to ...
sunghoon kim's user avatar
2 votes
0 answers
52 views

Deriving connection formulas for the incomplete elliptic integral of the third kind

For $\left(\varphi,\nu,\kappa\right)\in\mathbb{R}^{3}$ such that $-\frac{\pi}{2}\le\varphi\le\frac{\pi}{2}\land-1<\kappa<1\land0<1-\nu\sin^{2}{\left(\varphi\right)}$, the incomplete elliptic ...
David H's user avatar
  • 30.7k

1
2 3 4 5
14