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

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2
votes
1answer
93 views

Fourier serie of $\sqrt{1 - k^2 \sin^2{t}}$

I'm struggling with a Fourier serie. I need to find the Fourier series of the following function. That's the function under study: $f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$. The function ...
4
votes
1answer
70 views

Indefinite integral typo in Gradshteĭn: reciprocal square-root of sixth degree polynomial

The indefinite integral below, quoted from Gradshteĭn's [Table of Integrals, Series, and Products][1], 7th ed., appears to contain at least two typos (highlighted in purple). $\mathbf{2.291.4}$ ...
0
votes
0answers
33 views

Integrating the function $\alpha^{2}\sin^{2}\left( \dfrac {2 \pi x}{w}\right)$

I need to find $\int ^{x}_{0} \alpha^{2}\sin^{2}\left( \dfrac {2 \pi x}{w}\right) dL$ where $L=\dfrac {2kE\left( \dfrac {2\pi \alpha }{k}\right) }{\pi}\approx \dfrac {k+w}{2}\left( \dfrac {3\left( ...
4
votes
3answers
162 views

Integral of inverse of square root of quartic function with real roots

I was doing a physics problem and in order to finish it, I need to prove that: $\int_{x1}^{x2}\frac{dx}{{((x1 - x)(x - x2)(x - x3)(x - x4))}^{1/2}} = \int_{x3}^{x4}\frac{dx}{{((x1 - x)(x - x2)(x - ...
10
votes
3answers
185 views

How do I integrate $\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}$

How do I evaluate this indefinite integral, for $|k| < 1$: $$ \int\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}\mathrm{d}x $$ I tried the change of variable $t=\sin x$, and obtained two integrals, but I ...
0
votes
0answers
20 views

Is there a nice way to solve this Ramanujan-like approximation?

I have managed to approximate $l=\frac{2 \sqrt{4 \pi ^2 A^2+W^2} E(\epsilon )}{\pi }$ with parameter $\epsilon =k=\frac{2 \pi A}{\sqrt{4 \pi ^2 A^2+W^2}}$ alike Ramanujan as $l \approx ...
3
votes
1answer
54 views

Changes of variables to get an Elliptic Integral of the First Kind

I'm working with a non-linear second order ODE which has an analytical solution in terms of the Jacobi elliptical function $sn(u|k^2)$. The equation is $y''=y(\gamma - \frac{y^2}{2})$ where $\gamma$ ...
1
vote
0answers
73 views

Approximating an Infinite Summation into Closed Form

I want to be able to approximate the following in closed form $$L=\sqrt{\gamma ^2+4 \pi ^2 \psi ^2} \sum _{j=0}^{\infty } \frac{\left(\frac{(2 j-1)\text{!!} (2 \pi \psi )^j}{(2 j)\text{!!} ...
3
votes
0answers
47 views

How can I find my eccentricity (k, of the incomplete elliptic integral of the second kind) using a binomial series or root-finding algorithm

My main objective is to rearrange the following to find $A=$. I start with $C=\int^{B}_{0}\sqrt {1+\dfrac {A^{2}}{B^{2}}\cos^{2}\left(\dfrac {x}{B}\right) }dx$ By substituting $y=x/B$ and using ...
0
votes
0answers
44 views

Value of an elliptic integral of the first kind

The elliptic integral of the first kind $$ \int_0^{\pi/2}{\frac{du}{\sqrt{1-k^2\sin^2{u}}}} $$ cannot be expressed in terms of standard functions. But in the following context from The Pendulum by ...
0
votes
2answers
41 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
4
votes
2answers
124 views

An identity of an Elliptical Integral

Suppose $0<k<1$ and $\displaystyle K(k)=\int_0^1\frac{\mathrm{d}x}{\sqrt{(1-x^2)(1-k^2x^2)}}$. Let $\tilde{k}$ be $\tilde{k}^2=1-k^2$. Show that $$\displaystyle ...
4
votes
1answer
80 views

Elliptic integral $\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk$

Question: Prove that $$\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk=\frac{1}{16\pi}\Gamma^4\left( \frac{1}{4}\right)$$ My attempt Start by the transformation $$k \to \frac{2\sqrt{k}}{1+k}$$ ...
1
vote
1answer
30 views

Numerical evaluation of the first (K) and second (E) complete elliptic integrals

To get a numerical evaluation of the first (K) and second (E) complete elliptic integrals: $$K(k)=\int_0^1\frac{dt}{(1-t^2)^{1/2}(1-k^2t^2)^{1/2}}, \ \ \ \ \ ...
1
vote
1answer
29 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
3
votes
2answers
123 views

Elliptical Integrals

I was trying to figure out the length of the arc in a single cycle of a sinusoidal curve and I used the curve length formula to arrive at $$\int_0^{2\pi}\sqrt{1+\cos^2x}\ dx,$$ which I am fairly ...
2
votes
0answers
26 views

Expression of $2\int_{0}^{\frac{1}{r_{0}}}\frac{du}{\sqrt{\frac{r_{0}-r_{s}}{r_{0}^{3}}-u^{2}\left(1-u r_{s}\right)}}$ in terms of elliptic integrals

In Gravitation by Misner et al. 1973 the authors state that (calculus related to the Schwarzschild metric page 678) : ...
1
vote
0answers
53 views

Closed form of $\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1)$

Is there a known closed form of the series below? $$\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1]$$
4
votes
1answer
57 views

Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
3
votes
1answer
100 views

TaylorSeries of complete elliptic integral of the first kind

I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$. Can someone help me? $$ K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 sin^2t}} $$ Results so far: $$ ...
7
votes
0answers
136 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\left(\frac{1}{e^n}\right)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
10
votes
5answers
286 views
4
votes
1answer
131 views

Does $\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi $ have a closed form in terms of elliptic integrals?

Consider the following integral for real $a, b$ such that the square root is real: \begin{equation} I=\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi \end{equation} For $a = 0$, the integral is easily ...
1
vote
1answer
53 views

Infinite series representation of elliptic integrals $F(k,\phi)$ and $E(k,\phi).$

Here is my working so far. For a natural number $n,$$$2^{2n-1}\left(-1\right)^n\sin^{2n}\phi=\frac{\left(-1\right)^n}{2}\binom{2n}{n}+\sum\limits_{r=0}^{n-1}\binom{2n}{r}\left(-1\right)^r\cos ...
4
votes
4answers
233 views

Prove that these two curves have the same length

My midterms are approaching, and I was going through some of our past Calculus midterms when I stumbled upon this question from 1996: Show that these two curves, $$(\Gamma) : \frac ...
0
votes
1answer
60 views

How to get arc-length of polar function $r= 4(1-\sin{\phi})$?

How can I get arc-length of this polar function? $$ r= 4(1-\sin{\phi})$$ $$-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$$ I know that arc-length of polar function can get calculate by ...
0
votes
0answers
16 views

Relationship between elliptic integrals of first and second order.

A colleague and I are making an issue where order to reach the result we are asked to prove identity between elliptic integrals first and second order. This is: $$\frac{V}{\pi}\left( ...
14
votes
1answer
266 views

Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
4
votes
1answer
118 views

Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$

I'm interested in the following integral: $$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$ where $K(z)$ is the complete elliptic integral of the 1ˢᵗ ...
4
votes
1answer
106 views

Proving Legendres Relation for elliptic curves

The legendre's relation can be stated as follows $$ K(k) E(k^*)+ E(k) K(k^*) - K(k) K(k^*) = \frac{\pi}{2} $$ where $k^* = \sqrt{1 - k^2}$ is the complimentary modulus, and $E$ and $K$ are ...
10
votes
1answer
172 views

How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
12
votes
1answer
294 views

How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
1
vote
1answer
96 views

The transformations on the nome and Landen's transformation

Could someone please explain how to transform the nome $q = e^{-\pi K'/K}$ from $q^2$ to $q$ and then to $-q$? In other words, how does changing $q^2$ to $q$ and then $q$ to $-q$ affect $k$ and $K$. ...
0
votes
2answers
54 views

A general solution to the sine-gordon equation (with only time dependence)

I have a non-linear equation similar to the sine-gordon equation, but it is ordinary: $\frac{d^2x}{dt^2} - g \sin(x) = 0$ where $g$ is some positive constant. I'm looking for a general solution of ...
1
vote
0answers
30 views

Inverse of an arbitrary elliptic integral

Given an arbitrary elliptic integral $I\left( x \right) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt,$ where $R$ is a rational function of its arguments and $P\left( t \right)$ a polynomial ...
1
vote
0answers
50 views

Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} ...
0
votes
0answers
75 views

Solid angle of an off axis disk

In the diagram on the first page of this paper, what does the point c represent? Alternatively, what does the length R, the length r, or the angle theta represent? This is what I know so far. The ...
2
votes
0answers
74 views

Evaluation of $A$ in $2K(\sqrt{x}) = -\log(1 - x) + A + o(1)$ when $x \to 1^{-}$

Let $$K(k) = \int_{0}^{\pi/2}\frac{dx}{\sqrt{1 - k^{2}\sin^{2} x}}$$ be the complete elliptic integral of first kind where $0 < k < 1$. Let $k' = \sqrt{1 - k^{2}}$ be the complementary modulus. ...
5
votes
2answers
381 views

Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$

Is there a closed form expression for the definite integral $$I=\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$$ for $a<b<c<d$? Mathematica 9.0 can do it for special cases using ...
1
vote
1answer
274 views

Heuman Lambda Function in MATLAB

I'm trying to implement the Heuman Lambda function in MATLAB, but i'm having problems getting a correct answer. The Heuman Lambda Function is: $$ \Lambda_0(\beta,k) = \frac{2}{pi}[E(k)F(\beta,k') + ...
3
votes
1answer
836 views

How to compute elliptic integrals in MATLAB

I need to calculate the complete elliptic integrals of the first and second kind , the incomplete elliptic integral of the first kind, and the incomplete elliptic integral of the second kind in ...
7
votes
1answer
153 views

Understanding why $\int_0^{\pi/2} \sqrt{1+\cos^2x} \geq \frac{\pi}{4}\bigl( 1 + \sqrt{2}\bigr)$

Lately I stumbled accros the magnifient paper by Roger Nelsen, which can be found here Symmetry and Integration In this paper it is shown that $$ \int_0^{\pi/2} \frac{\mathrm{d}x}{1 + ...
2
votes
0answers
71 views

How to compute the arc length of $f(x) = ax + b \sin(x)$.

I would like to compute the length of the arc of $f(x) = a x + b \sin(x)$ (let's say from $0$ to $\alpha < 2\pi$.). The traditional method of computing it as the integral $\int_0^\alpha ...
0
votes
1answer
274 views

Jacobian of an ellipse

An ellipse is given by $$ \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$$ You want to find the area by using a change of coordinates: $x = r\cos θ$, $y = \frac{br}{a}\sin θ$. Find the range of values of ...
25
votes
1answer
313 views

Conjectured closed form for $\int_0^1x^{2\,q-1}\,K(x)^2dx$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind

I am interested in a general closed-form formula for integrals of the following form: $$\mathcal{J}_q=\int_0^1x^{2\,q-1}\,K(x)^2dx,\tag0$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ ...
0
votes
2answers
60 views

Elliptic Integrals of the First Kind

Suppose I have $$F(\phi(x), k) = x$$ where the elliptic integral of the first kind is defined to be $$F(\phi, k) = \int_{0}^{\phi} \frac{1}{\sqrt{1-k^2\sin(\theta)}} \, d\theta $$ How could I invert ...
8
votes
1answer
274 views

Indefinite integral $\int \arcsin \left(k\sin x\right) dx$

It would take too long to explain the context reasonably well - but in short, this integral, or rather its equivalent $$\int\frac{x\cos x\,dx}{\sqrt{1-k^2\sin^2x}},\qquad 0<k<1$$ is related to ...
7
votes
0answers
134 views

Closed-form expression for an iterated integral

Does the following iterated integral have a simple closed-form expression in terms of $z$? $$ I = \int_0^\infty \int_0^\infty \sqrt{\frac{1 + x^2 y^2 + x^2 z^2 + y^2 z^2}{(x^2 + y^2 + z^2 + x^2 y^2 ...
1
vote
1answer
68 views

how do i give upper and lower limits to an elliptical integral---

how do i give upper and lower limits to an elliptical integral? I want to compute: $$\int_0^{2\pi} \frac{1}{\sqrt{1-k^2\sin^2x}} dx$$ for which I get $F(x/k^2)$ I need to put upper and lower ...
11
votes
2answers
263 views

Show that $K=K'$ if and only if $k=(\sqrt{2}-1)^2$ (Ahlfors)

In Ahlfors' Complex Analysis text, page 240, he defines the following two integrals: $$K=\int_{-1}^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}}, $$ $$K'=\int_1^{1/k} \frac{dt}{\sqrt{(t^2-1)(1-k^2t^2)}}. $$ ...