New posts in elliptic-integrals

Does $\int_0^{2\pi} \sqrt{(1-a\cos x)^2 + b} \mathrm{d}x$ evaluate in terms of elliptic integrals?

integration definite-integrals indefinite-integrals elliptic-integrals

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

integration special-functions elliptic-integrals

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

integration definite-integrals elliptic-integrals

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

calculus real-analysis sequences-and-series complex-analysis elliptic-integrals

Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$

calculus integration definite-integrals closed-form elliptic-integrals

A logarithmic integral, generalization of a result of Shalev

definite-integrals summation special-functions elliptic-integrals polylogarithm

Evaluating the elliptic integral $\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}}$

special-functions definite-integrals elliptic-integrals

Can $\Gamma(1/5)$ be written in this form?

integration special-functions gamma-function elliptic-integrals

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

integration improper-integrals special-functions lambert-w elliptic-integrals

Derivative of the elliptic integral of the first kind

calculus special-functions elliptic-integrals

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

integration definite-integrals special-functions elliptic-integrals

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

integration special-functions elliptic-integrals

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

complex-analysis integration improper-integrals elliptic-integrals

Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$

improper-integrals gamma-function closed-form elliptic-integrals

An identity of an Elliptical Integral

integration elliptic-integrals

Origin of rapidly converging sequence for $e^\pi$

sequences-and-series elliptic-integrals experimental-mathematics

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

integration definite-integrals special-functions closed-form elliptic-integrals

What even *are* elliptic functions?

complex-analysis special-functions analytic-number-theory elliptic-integrals elliptic-functions

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

integration analysis definite-integrals closed-form elliptic-integrals

Integral with arithmetic-geometric mean ${\large\int}_0^1\frac{x^z}{\operatorname{agm}(1,\,x)}dx$

calculus integration definite-integrals closed-form elliptic-integrals