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Questions tagged [elliptic-integrals]

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

2
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3answers
139 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 \...
1
vote
2answers
44 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 ...
11
votes
1answer
217 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 ...
0
votes
0answers
41 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 ...
0
votes
0answers
20 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) = ...
0
votes
2answers
50 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:$$K^{-1}(K(k))=k$$
14
votes
2answers
203 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}}$...
4
votes
1answer
149 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
votes
0answers
23 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
votes
1answer
69 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
votes
0answers
35 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
votes
0answers
42 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
votes
1answer
63 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
vote
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 ...
0
votes
0answers
32 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
votes
1answer
72 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
votes
3answers
126 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'...
4
votes
0answers
66 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=...
0
votes
1answer
37 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 ...
7
votes
1answer
112 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}\...
0
votes
2answers
59 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 ...
0
votes
1answer
89 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 ...
0
votes
0answers
22 views

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 ...
1
vote
2answers
184 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
votes
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
votes
1answer
103 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\...
2
votes
1answer
110 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 ...
1
vote
0answers
55 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)}(\...
0
votes
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
78 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
votes
0answers
34 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
237 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 ...
-2
votes
2answers
125 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
vote
1answer
86 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
vote
1answer
70 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
40 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
52 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 ...
1
vote
2answers
109 views

Can we write elliptic integrals to integrate arbitrary 4th degree polynomial as a programming function?

I have try to solve the problem about the length of cubic bezier curve, in general it is cubic polynomial function $a_pt^3 + b_pt^2 + c_pt + d_p$ I think the method is differentiate this function ...
3
votes
0answers
59 views

Elliptic Integral from Gradshteyn and Ryzhik

In my course of work I came across the following elliptic integral, and found its solution in Gradshteyn and Ryzhik: $$\int_{b}^{u} \frac{\mathrm{d} x}{\sqrt{(a-x)(x-b)(x-c)(x-d)}} = \frac{2}{\sqrt{(...
2
votes
1answer
79 views

Conversion to elliptic integrals

I have the following differential equation: $$2\theta'' = h \sin\theta$$ After multiplying both sides by $\frac{\mathrm{d}\theta}{\mathrm{d}x}$, I obtain: $$\frac{\mathrm{d}}{\mathrm{d}x} \left[\...
1
vote
0answers
28 views

Rationality of circumference of an ellipsis with rational semi-axes

We all know that the ratio of circumference of a circle to the radius is a transcendental number, but how about ellipses? It is well known that the circumference of an ellipse with semi-axes lengths $...
3
votes
2answers
166 views

Asymptotic expansion of complete elliptic integral of third kind

Is there a way to compute the expansion of the complete elliptic integral of third kind $\Pi(n,k)=\int_0^{\pi/2} \frac{d\varphi}{(1-n\sin^2\varphi)\sqrt{1-k^2\sin^2\varphi}}$ for $\Pi(1+\epsilon,1-\...
0
votes
2answers
40 views

What is the relation in terms of the integral $\int_{b}^{a}(cos( b)-cos( x))^\frac{1}{2}dx$?

I am considering the relation between the integral$$I_1=\int_{b}^{a}(cos( b)-cos( u))^\frac{1}{2}du$$ and $$I_2=\int_{b}^{a}(cos(k\cdot b)-cos(k\cdot v))^\frac{1}{2}dv$$where $a>b>0$ and $k>0$...
0
votes
0answers
43 views

How to transfer integral to elliptic one

I have the following function $$E(\theta)=\int \frac{d\theta}{\sqrt{(1-\alpha\cos\theta)(1+\beta\cos\theta)}},$$ I cannot find a suitable change of variable such that $E(\theta)$ converted into ...
5
votes
2answers
219 views

On the asymptotic behavior of the Fourier coefficients of $(1-R\cos\theta)^{-3/2}$

Today in class I showed some ways for dealing with the classical integral $\int_{0}^{2\pi}\frac{d\theta}{(A+B\cos\theta)^2}$ under the constraints $A>B>0$, including Symmetry and the tangent ...
1
vote
0answers
51 views

elliptic integral - gauss theorem

In order to proove Gauss theorem on elliptic integrals it's written : For any couples $(a, b$), let us defines : $$\displaystyle \mathcal{T}(a, b) = \int_{0}^{+\infty} \frac{1}{\sqrt{(a^2+t^2)\cdot(b^...
3
votes
3answers
231 views

True value of $\int_0^{\pi/2}\frac{dx}{(1-2\sin^2 x)\sqrt{1-4\sin^2 x}}$

During the generation of test values I stumbled over this question What is the true value of the complete elliptic integral of the 3rd kind $$I=\int_0^{\pi/2}\frac{dx}{(1-2\sin^2 x)\sqrt{1-4\sin^...
0
votes
0answers
40 views

A simple, compact, reasonably accurate interpolation for the complete elliptic integral of the second kind.

A simple, compact, reasonably accurate, interpolation for the complete elliptic integral of the second kind, which I felt I would share: In the context of a study of percolation in systems of ...
1
vote
1answer
58 views

Can this integral be reduced to an elliptic integral, or to anything else that is easier to evaluate?

New user here......I have run into a problem where I am trying to evaluate the following integral (if at all possible, analytically): $ I = \int_{0}^{\pi/2}\sin(x) \sqrt{1 + k^2 \sin^2(x)}\, dx $ ...
1
vote
0answers
132 views

Jacobi form to Weierstrass form . . . lattices included … polynomial factoring in the way

Slightly more meat edit : I want to solve this integral $$ J_{k=1} = \int dt \sqrt(2 t^6 - 2 t^5 - t^4 + t^2 + 1) - 1 $$ I know about some approximation schemes, but I want to grow bigger so I am ...