Questions tagged [elliptic-integrals]
Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.
0
votes
1answer
31 views
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
votes
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 ...
0
votes
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
votes
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 ...
1
vote
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 ...
0
votes
0answers
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
votes
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 ...
2
votes
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) ...
1
vote
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 ...
9
votes
0answers
152 views
About the product of two Elliptic integrals
Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...
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votes
2answers
52 views
Asymptotic expression for the complete Elliptic integral of the first kind
On the Wikipedia page Elliptic integral it states, that the complete elliptic integral of the first kind has asymptotic expression
$$K(k) = \frac{\pi}{2}+\frac{\pi}{8}\left(\frac{k^2}{1-k^2}\right)-\...
2
votes
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:
$$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(x))}$$...
0
votes
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 ...
3
votes
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
$$ \operatorname{F}(\varphi,k) = \int \limits_0^\varphi \frac{\mathrm{d} \theta}{\sqrt{1-k^2 \sin^2(\theta)}} = \int \...
1
vote
2answers
53 views
How to reduce an integral with square root of cubic function into an elliptic integral
I need to calculate the following ntegral:
$$\int \frac{t }{\sqrt{2 t^3 - 3 t^2 + 6 C}} dt$$
where $C$ is a constant to be determined later, so I cannot look for roots of the polynomial in the ...
11
votes
1answer
249 views
Integral $\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}$
How can we prove $$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)?$$
Thoughts of this integral
This ...
0
votes
0answers
47 views
Expressing the solution of $\int\frac{x^2 dx}{a-\sqrt{b^2+x^2}-\sqrt{c^2+x^2}}$ without the use of elliptic integrals of complex arguments?
Mathematica is able to provide me with an analytic solution to this integral, but it involves EllipticF's, EllipticE's, and EllipticPi's, all of complex variables. Are there any integral tables that ...
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0answers
35 views
Are there any identities for incomplete elliptic integrals of the third kind with complex arguments?
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $\tan\theta = \sinh\phi$
17.4.8
$F(i\phi | \alpha) = ...
0
votes
2answers
61 views
Numerical Calculation for Inverse Complete Elliptic Integral of The First Kind? [closed]
Is there a way to calculate the inverse of $K(k)$ which is the complete elliptic integral of the first kind.
Ex:$$K^{-1}(K(k))=k$$
14
votes
2answers
208 views
Closed form of an improper integral to solve the period of a dynamical system
This improper integral comes from a problem of periodic orbit. The integral evaluates one half of the period.
In a special case, the integral is
$$I=\int_{r_1}^{r_2}\frac{dr}{r\sqrt{\Phi^2(r,r_1)-1}}$...
4
votes
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
votes
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
votes
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
votes
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
$$F(\phi,k):=\int_0^\phi \frac{dx}{\sqrt{1-k^2\...
3
votes
0answers
49 views
Complete elliptic integral $K(k) $ for $k>1$
I have always tried to work with elliptic integrals with modulus $k\in(0,1)$ to avoid the issues related to complex variables.
In what follows I have tried to link the integral of modulus greater ...
2
votes
1answer
67 views
Evaluate $\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1+k^2 \sin^2 \theta}} \,d \theta$
Evaluate $\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1+k^2 \sin^2\theta}}
\,d \theta$
I wang to let $k=-ai \,\,\,\,\,$ ,then :$$\int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1-a^2 \sin^2\theta}}
\,d \theta$$
...
1
vote
1answer
33 views
Integer solutions to AGM iteration
Any integer solution to $a^2+b^2=c^2$ also provides an integer solution $x=c$, $y=a$, $z=c+b$, $w=c-b$ to $$agm(x,y)=agm(z,w)$$ where $agm$ denotes Gauss' arithmetic-geometric mean. Are there other ...
0
votes
0answers
37 views
How to derive the volume of the region left when a plane cuts a solid elliptical cylinder?
I need to be able to calculate the volume of an elliptical cylinder that is cut by a plane. It is similar to slicing a cylinder and finding the volume, only this time with an elliptical cylinder. I ...
2
votes
1answer
77 views
Showing that $\sin x\;f(\sin x)\;f^\prime(\cos x)+\cos x\;f(\cos x)\;f^\prime(\sin x)=\frac{2}{\pi\sin x\cos x}$ for $f(x)$ defined by a series
Let
$$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
votes
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'...
4
votes
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 :
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow y=...
0
votes
1answer
41 views
Quotient of quarter periods K' and K of Jacobi elliptic functions
There are several ways to express the quarter period $K$,
$$
K(m)=\int_0^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{1-m\sin^2\theta}},
$$
as a power series (and thus for $K'=K(1-m)$ there are, too) and also ...
7
votes
1answer
122 views
Evaluating $K\big(\frac{3-\sqrt{7}}{4\sqrt{2}}\big)$
On MSE, I have seen derivations of the elliptic integral special values
$$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}\...
0
votes
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 ...
0
votes
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 ...
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
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.
$$L \approx \pi(a+b) \frac{(64-3d^4)}{(64-16d^2 )},\quad ...
3
votes
0answers
40 views
Precision on computation of the complete elliptic integrals
I am looking into elliptic integrals and I am slightly confused about something. The material I'm using for reference is mainly Wikipedia and Higher Transcendental Functions II (page 306 onward).
The ...
7
votes
1answer
107 views
How to integrate $\int\frac{1}{\sqrt{x(x-9)(x-5)}}\,dx$?
Integrate:
$$\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\...
2
votes
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 ...
1
vote
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)}(\...
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
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
votes
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)$$
where $q=e^{2\pi i \tau}$ is ...
-2
votes
2answers
136 views
Definite Integral: $\int_{0}^{2\pi}\frac{d \theta }{\sqrt{1-k^{2}\cos( \theta )}}$
I need integral result for following integral:
$$\int_{0}^{2\pi}\frac{d \theta }{\sqrt{1-k^{2}\cos( \theta )}}$$
It will be useful in an electromagnetic simulator. It is obtained as the medium 1/...
1
vote
1answer
120 views
Real and imaginary parts for incomplete elliptic integral 1st kind with complex argument
I have this case: $F(\arcsin u,m)$, where $u>1$, so it ends up as $F(\frac{\pi}{2}-i\text{arccosh} u,m)$.
and I need to decompose it into real and imaginary parts. I have seen this question, but ...
1
vote
1answer
72 views
Evaluate in terms of elliptic integral
Define
$$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 ...
