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Questions tagged [elliptic-integrals]

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

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How to find coordinates that split this arc into 3 pieces of equal length?

I need the $x$ coordinates of two new points that split an elliptical arc between 2 given points in quadrant I into 3 equal pieces. How can I get these coordinates? I want to use Wolfram but I can ...
5
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1answer
73 views

Addition formula for elliptic integral of second kind

Let $k\in(0,1)$ and the incomplete elliptic integral integral $E(u, k) $ be defined by $$E(u, k) =\int_{0}^{u}\operatorname {dn} ^2(t,k)\,dt\tag{1}$$ where $\operatorname {dn} (u, k) $ represents one ...
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0answers
10 views

Is there a simple theorem about how to represent any function defined by a integral as a Taylor Series around a specific point?

Suppose I have a function $f$ defined as $f(x)=\int^{x}_{a}(L(t))dt $ where $L(t)$ is a known function, yet $f(x)$ can’t be represented with elementary functions, such as an elliptic integral, and yet ...
2
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2answers
61 views

Complete Elliptic Integral of the First Kind Identity

Is there an identity for $\frac{K'(k)}{K(k)}=?$ where $K(k)=\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{1-k^2\sin^2(x)}}dx=\int_0^1\frac{1}{\sqrt{(1-t^2)(1-k^2t^2)}}dt$ is the Complete Elliptic Integral of ...
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0answers
31 views

The fundamental solution for Laplace's equation in cylindrical coordinates

I am working my way through the excellent textbook of Garabedian on partial differential equations and have two questions related to the topic of the fundamental solution in chapter 5: Garabedian ...
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10 views

Plotting incomplete elliptic integral of 1st kind

The incomplete elliptic integral of 1st kind is: $$F(\varphi,k)=\int_{0}^{\varphi} \frac{1}{\sqrt{1 - k^2 \sin^2(x)}} \mathrm{d}x$$ I wanted to set a small dataframe in order to plot myself some ...
13
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1answer
288 views

A logarithmic integral, generalization of a result of Shalev

As many of you are already aware, I and Marco Cantarini are currently working on the applications of fractional operators to hypergeometric series, extending the class of $\phantom{}_{p+1} F_p$s whose ...
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2answers
30 views

Concrete example using elliptic integral of the second kind to calculate arc length?

There's an identical question here but it was never answered fully and the link providing an essential component of the accepted "answer" is broken. The Keisan website presents a solution here (PDF) ...
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1answer
25 views

Incomplete elliptic integral and Jacobi's form

The incomplete elliptic integral of the first kind is written (using trigonometric form) : $ F(\varphi,k)=\int_{0}^{\varphi} \frac{1}{\sqrt{1-k^2 \sin^2(\theta)}} \mathrm{d}\theta $. Then, it is ...
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0answers
152 views

About the product of two Elliptic integrals

Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...
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2answers
52 views

Asymptotic expression for the complete Elliptic integral of the first kind

On the Wikipedia page Elliptic integral it states, that the complete elliptic integral of the first kind has asymptotic expression $$K(k) = \frac{\pi}{2}+\frac{\pi}{8}\left(\frac{k^2}{1-k^2}\right)-\...
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0answers
57 views

Hyperelliptic function addition formula

$$x= \int_{0}^{f(x)} \frac{du}{\sqrt{1-u^2}\sqrt{1-k^2u^2}\sqrt{1-l^2u^2}}$$ $$f(0)=0$$ If we apply derivative operation for both sides, we get: $$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(x))}$$...
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1answer
65 views

How do I integrate the equation of a ellipse?

How do I integrate this equation with respect to "x"? I am quite sure I need to use trigonometric substitution (or even hyperbolic), but I am not good at these methods since I just learned about ...
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3answers
166 views

Asymptotic expansion of $\int_0^1 \frac{\operatorname{K}(r x)}{\sqrt{(1-r^2 x^2)(1-x^2)}} \, \mathrm{d} x $

Notation: For $\varphi \in [0,\frac{\pi}{2}]$ and $k \in [0,1)$ the definitions $$ \operatorname{F}(\varphi,k) = \int \limits_0^\varphi \frac{\mathrm{d} \theta}{\sqrt{1-k^2 \sin^2(\theta)}} = \int \...
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2answers
53 views

How to reduce an integral with square root of cubic function into an elliptic integral

I need to calculate the following ntegral: $$\int \frac{t }{\sqrt{2 t^3 - 3 t^2 + 6 C}} dt$$ where $C$ is a constant to be determined later, so I cannot look for roots of the polynomial in the ...
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1answer
249 views

Integral $\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}$

How can we prove $$I:=\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}\\=\frac1{6\sqrt2\pi^2}\Gamma(1/11)\Gamma(3/11)\Gamma(4/11)\Gamma(5/11)\Gamma(9/11)?$$ Thoughts of this integral This ...
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0answers
47 views

Expressing the solution of $\int\frac{x^2 dx}{a-\sqrt{b^2+x^2}-\sqrt{c^2+x^2}}$ without the use of elliptic integrals of complex arguments?

Mathematica is able to provide me with an analytic solution to this integral, but it involves EllipticF's, EllipticE's, and EllipticPi's, all of complex variables. Are there any integral tables that ...
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0answers
35 views

Are there any identities for incomplete elliptic integrals of the third kind with complex arguments?

Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments: For $\tan\theta = \sinh\phi$ 17.4.8 $F(i\phi | \alpha) = ...
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2answers
61 views

Numerical Calculation for Inverse Complete Elliptic Integral of The First Kind? [closed]

Is there a way to calculate the inverse of $K(k)$ which is the complete elliptic integral of the first kind. Ex:$$K^{-1}(K(k))=k$$
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2answers
208 views

Closed form of an improper integral to solve the period of a dynamical system

This improper integral comes from a problem of periodic orbit. The integral evaluates one half of the period. In a special case, the integral is $$I=\int_{r_1}^{r_2}\frac{dr}{r\sqrt{\Phi^2(r,r_1)-1}}$...
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1answer
152 views

Evaluating the integral $\int_0^{\pi/4}\sqrt{1-16\sin^2(x)}\mathop{}\!\mathrm dx$

How can we evaluate this integral? $$\int_\limits{0}^{\pi/4}\sqrt{1-16\sin^2(x)}\mathop{}\!\mathrm dx$$ I tried a substitution $$u=4\sin x,\quad \mathrm dx=\frac{\mathrm du}{\sqrt{16-u^2}}$$ hence ...
2
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0answers
28 views

How can the following be transformed in to a sum of complete elliptic integrals of the first and second kind

I have the following, that I known from a numerical implementation of the problem by a third party should be able to be transformed in to elliptic integrals of the first and second kind however I can'...
2
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1answer
77 views

Solving the nonlinear differential equation $ m \ddot x +\alpha x + \beta x^3 = 0$

As the header says: I want to solve the differential equation $ m \ddot x +\alpha x + \beta x^3 = 0$, with initial conditions $x(0) = -x_0$, $\dot x(0)=0$. It comes up in the solution to the equations ...
2
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0answers
49 views

Addition formula for $\text{sn}(u)=\text{sn}(u,k)$

I have just learned the definition of the first Jacobian elliptic function $\text{sn}(u)=\text{sn}(u,k)$, defined as the sine of the inverse function of $$F(\phi,k):=\int_0^\phi \frac{dx}{\sqrt{1-k^2\...
3
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0answers
49 views

Complete elliptic integral $K(k) $ for $k>1$

I have always tried to work with elliptic integrals with modulus $k\in(0,1)$ to avoid the issues related to complex variables. In what follows I have tried to link the integral of modulus greater ...
2
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1answer
67 views

Evaluate $\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1+k^2 \sin^2 \theta}} \,d \theta$

Evaluate $\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1+k^2 \sin^2\theta}} \,d \theta$ I wang to let $k=-ai \,\,\,\,\,$ ,then :$$\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1-a^2 \sin^2\theta}} \,d \theta$$ ...
1
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1answer
33 views

Integer solutions to AGM iteration

Any integer solution to $a^2+b^2=c^2$ also provides an integer solution $x=c$, $y=a$, $z=c+b$, $w=c-b$ to $$agm(x,y)=agm(z,w)$$ where $agm$ denotes Gauss' arithmetic-geometric mean. Are there other ...
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0answers
37 views

How to derive the volume of the region left when a plane cuts a solid elliptical cylinder?

I need to be able to calculate the volume of an elliptical cylinder that is cut by a plane. It is similar to slicing a cylinder and finding the volume, only this time with an elliptical cylinder. I ...
2
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1answer
77 views

Showing that $\sin x\;f(\sin x)\;f^\prime(\cos x)+\cos x\;f(\cos x)\;f^\prime(\sin x)=\frac{2}{\pi\sin x\cos x}$ for $f(x)$ defined by a series

Let $$f(x) = 1 + \left(\frac12\cdot x\right)^2+\left(\frac12\cdot\frac34 \cdot x^2\right)^2+\left(\frac12\cdot\frac34\cdot\frac56\cdot x^3\right)^2+\cdots$$ Prove that $$\sin x\;f(\sin x)\;f^\...
4
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3answers
129 views

How to evaluate the integral: $\int_0^\frac{\pi}{2} \cos(x)\sqrt{\cos(x)} \,dx$

$$\int_0^\frac{\pi}{2} \cos(x)\sqrt{\cos(x)} \,dx$$ I've been trying to find a way to integrate this function for a while. From my research I think this should reduce to an elliptic integral but I can'...
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0answers
76 views

Exact value of Elliptic Integrals.

I was taking currently in a elementary calculus course where i found how to find arc lengths of a smooth continuous curve. so here is how i started : $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow y=...
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1answer
41 views

Quotient of quarter periods K' and K of Jacobi elliptic functions

There are several ways to express the quarter period $K$, $$ K(m)=\int_0^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{1-m\sin^2\theta}}, $$ as a power series (and thus for $K'=K(1-m)$ there are, too) and also ...
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1answer
122 views

Evaluating $K\big(\frac{3-\sqrt{7}}{4\sqrt{2}}\big)$

On MSE, I have seen derivations of the elliptic integral special values $$K(1/\sqrt{2})=\frac{\Gamma^2(1/4)}{4\sqrt{\pi}}$$ $$K(\tan(\pi/8))=\frac{\sqrt{\sqrt{2} +1} \Gamma (1/8)\Gamma (3/8)}{2^{13/4}\...
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2answers
65 views

Solving for argument of complete elliptic integral of first kind

I have the following equation to be solved for $m$ $\frac{K(1-m)}{K(m)} = a$ where $a$ is known value, $K(m)$ is the complete elliptic integral of the first kind. $K(m)$ can be expressed as an ...
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1answer
91 views

Calculate $\int \frac{r \: dr}{\sqrt{r^4 l^{-2}+ r^2(1 + a^2 l^{-2}) - 2 m r + a^2}}$

How to solve this integral? $\displaystyle \int \frac{r \: dr}{\sqrt{r^4 l^{-2}+ r^2(1 + a^2 l^{-2}) - 2 m r + a^2}}$ where, $a,l,m \in \mathbb{R}$ and $m > 0$. I tried putting this in ...
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0answers
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Use Taylor series to calculate C(a,2a)/ C(a,a) accurate to 0.001

I have done 1 and 2. I think C(a,a) is a circle. I want to do No 4. I have got a answer to no 3 for E. But I havent idea to do no 4. I have got a answer to C(a,2a)/ C(a,a) from E. After that what how ...
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2answers
311 views

What is the fastest way to estimate the Arc Length of an Ellipse?

To estimate the circumference of an ellipse there are some good approximations. $a$ is the semi-major radius and $b$ is the semi-minor radius. $$L \approx \pi(a+b) \frac{(64-3d^4)}{(64-16d^2 )},\quad ...
3
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0answers
40 views

Precision on computation of the complete elliptic integrals

I am looking into elliptic integrals and I am slightly confused about something. The material I'm using for reference is mainly Wikipedia and Higher Transcendental Functions II (page 306 onward). The ...
7
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1answer
107 views

How to integrate $\int\frac{1}{\sqrt{x(x-9)(x-5)}}\,dx$?

Integrate: $$\int\frac{1}{\sqrt{x(x-9)(x-5)}}\,dx$$ I did some substitutions, but it seems not to be the right path to follow. Some hints? Noticing that $x(x-9)(x-5) =x((x-7)^2-4)$ we have: $$\int\...
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1answer
115 views

Evaluate $\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{a^2 \cos^2 \theta+b^2 \sin^2 \theta}} d \theta$

$\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{a^2 \cos^2 \theta+b^2 \sin^2 \theta}} d \theta$ $ = \int_0^{\frac{\pi}{2}} \frac{1}{a}\sec \theta \frac{1}{ \sqrt{1+(b/a)^2 \tan^2 \theta}} d \theta$ But i ...
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0answers
58 views

Multiple integral involving trigonometric functions over a hypercube.

Let $d\ge 1$ be an integer and $\vec{A} := \left(A_j\right)_{j=1}^d \in {\mathbb R}^d$ subject to $\sum\limits_{j=1}^d A_j^2 \le 1$. Define a following integral: \begin{equation} {\mathfrak I}^{(d)}(\...
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1answer
88 views

Integral of $\int\frac{\mathrm dx}{\sqrt{a+bx^{2\left(1-\frac{1}{k}\right)}-x^2}}$

Can you do $$\int\frac{\mathrm dx}{\sqrt{a+bx^{2\left(1-\frac{1}{k}\right)}-x^2}}$$ with $a,b\geq0$ and $k$ is a integer bigger than $1$. Can it be expressed in terms of elliptic integrals? Most of ...
1
vote
1answer
83 views

Converting a function of $\cos^2$ to a complete elliptic integral of the first kind

I am having a hard time following Equation 2.6 of Taib, Bachok Bin. "Boundary integral method applied to cavitation bubble dynamics." (1985). The equation is on the middle of page 8 of the document or ...
0
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0answers
37 views

An identity involving complete elliptic integrals of the first and third kind.

Let $m = 1-k_3^2 = (2+\sqrt{3})/4$, where $k_3 = (\sqrt{6} - \sqrt{2}) / 4$ is the third elliptic singular value. So $K(m)/K'(m)=\sqrt{3}$. For this special value of $m$, Mathematica tells me that $$...
8
votes
4answers
267 views

How to derive relationship between Dedekind's $\eta$ function and $\Gamma(\frac{1}{4})$

I am trying to determine in what way to approach finding a connection between Dedekind's Eta Function, defined as $$\eta(\tau)=q^\frac{1}{24}\prod_{n=1}^\infty(1-q^n)$$ where $q=e^{2\pi i \tau}$ is ...
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2answers
136 views

Definite Integral: $\int_{0}^{2\pi}\frac{d \theta }{\sqrt{1-k^{2}\cos( \theta )}}$

I need integral result for following integral: $$\int_{0}^{2\pi}\frac{d \theta }{\sqrt{1-k^{2}\cos( \theta )}}$$ It will be useful in an electromagnetic simulator. It is obtained as the medium 1/...
1
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1answer
120 views

Real and imaginary parts for incomplete elliptic integral 1st kind with complex argument

I have this case: $F(\arcsin u,m)$, where $u>1$, so it ends up as $F(\frac{\pi}{2}-i\text{arccosh} u,m)$. and I need to decompose it into real and imaginary parts. I have seen this question, but ...
1
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1answer
72 views

Evaluate in terms of elliptic integral

Define $$f(t):=\int_0^1 \frac{\sqrt{x+t+\sqrt{t^2+2tx+1}}}{\sqrt{1-x^2}} dx $$ Can this integral be evaluated in terms of elliptic integrals? I ask because I have established the functional equation ...
0
votes
0answers
44 views

An identity for the complete elliptic integral of the first kind

On the Wolfram webpage, one can find the following identity for the complete elliptic integal of the first kind $K(z)$: $$K(z) = \frac{2}{1+\sqrt{1-z}} K\Big( \Big( \frac{1-\sqrt{1-z}}{1+\sqrt{1-z}} \...
2
votes
1answer
60 views

Calculating the lattice of the tori of a non-singular projective cubic curve

If $C$ is the curve in $\mathbb{C}\mathbb{P}^{2}$ defined by the zero set of the polynomial $P^{\lambda}(x,y,z) = y^{2}z - x(x-z)(x-\lambda z)$, for $\lambda$ not $0$ or $1$. Then we know that $C$ is ...