# Tagged Questions

69 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 $$k \to \frac{2\sqrt{k}}{1+k}$$ ...
49 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 ...
123 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: $$I=\int_0^{2 \pi} \sqrt{1-(a+b \sin\phi)^2} d\phi$$ For $a = 0$, the integral is easily ...
103 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 ...
92 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 ...
162 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: ...
286 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 ...
49 views

In reading these notes (elliptic curves starting from elliptic integrals) I came across a couple claims about the topology of some complex surfaces. On page 4, they discuss the integral $$\phi(x) = ... 2answers 358 views ### Evaluating the elliptic integral \int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}} I have the following integral,$$I(t)=\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}},$$where t>4 is a real parameter. I know from messing around numerically and playing with Mathematica that ... 2answers 1k views ### Is this function decreasing on (0,1)? While doing some research I got stuck trying to prove that the following function is decreasing$$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$for k \in (0,1). ... 2answers 265 views ### complete elliptic integral of the first kind I'm looking for any "closed form" for the coefficient of the \ell-th power of K(x), the complete elliptic integral of the first kind. Thanks. 2answers 308 views ### Writing complete elliptic integral of first kind as a hypergeometric function I am trying to show K=\int_{0}^{\frac{\pi}{2}}\frac{dz}{\sqrt{1-k^{2}\sin^{2}(z)}} can be written as \frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2}). First I used ... 1answer 316 views ### Identity for complete elliptic integral of the second kind Why does |x-1| \; E\left(-\dfrac{4x}{(x-1)^2}\right) = |x+1| \; E\left(\dfrac{4x}{(x+1)^2}\right), where E(m) is the complete elliptic integral of the second kind, with parameter m? 0answers 334 views ### Integral involving Complete Elliptic Integral of the First Kind K(k) I have run into an integral involving the complete elliptic integral, which can be put into the following form after changing integration variables to the modulus: ... 3answers 429 views ### Can this integral \int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}} be written in the form of a elliptic integral I am trying to find the magnetic field due to an elliptic loop of wire. How to do integrals of the type$$\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}} Where a and b are ...
The solid angle spanned by a disc of unit radius, as observed from a point $(r,z)$ at a distance $z>0$ above a point in the disc plane with at distance $r>0$ to the center, can be expressed as ...