New posts in closed-form

Closed-form analytical solutions to Optimal Transport/Wasserstein distance

probability statistics normal-distribution closed-form optimal-transport

Evaluating $\int_0^{\Large\frac{\pi}{2}}\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan(x)}\right)^3dx$

integration definite-integrals logarithms closed-form polygamma

What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

calculus integration definite-integrals logarithms closed-form

On the evalution of an infinite sum

integration sequences-and-series definite-integrals improper-integrals closed-form

A closed form for the sum $\frac{a}{b}+\frac{a\cdot(a+1)}{b\cdot(b+1)}+\frac{a\cdot(a+1)\cdot(a+2)}{b\cdot(b+1)\cdot(b+2)}+\cdots$

sequences-and-series binomial-coefficients closed-form fractions

How to check if some equation can be solved using Lambert $\operatorname{W}$ function.

closed-form lambert-w

Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

sequences-and-series closed-form harmonic-numbers

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$

calculus integration definite-integrals closed-form bessel-functions

Closed form for $\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx$

calculus definite-integrals closed-form conjectures hypergeometric-function

Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$

calculus complex-analysis closed-form conjectures polylogarithm

integral of $\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$

calculus real-analysis integration improper-integrals closed-form

What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$?

integration definite-integrals logarithms closed-form

Integral $\int_{0}^1\frac{\ln\frac{3+x}{3-x}}{\sqrt{x(1-x)}}dx$

calculus integration definite-integrals improper-integrals closed-form

Summation Theorem how to get formula for exponent greater than 3

algebra-precalculus summation closed-form faq

For $a>0$, what is $\int_1^\infty dy \, \exp(-ay) (y^2-1)^{-1/2}y^{-2}$?

definite-integrals special-functions closed-form bessel-functions

Find a function for the infinite sum $\sum_{n=0}^\infty \frac{n}{n+1}x^n$

calculus sequences-and-series taylor-expansion closed-form

Closed form (or an ODE) for the integral $\int_0^\infty \frac{1+z^2}{1+z^4} \frac{z^p}{1+z^{2p}} dz$

integration ordinary-differential-equations definite-integrals closed-form gamma-function

A closed form of $\sum_{n=1}^\infty\left[ H_n^2-\left(\ln n+\gamma+\frac1{2n} \right)^2\right]$

integration sequences-and-series definite-integrals closed-form harmonic-numbers

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$? [closed]

calculus sequences-and-series trigonometry contest-math closed-form

How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$

calculus special-functions closed-form conjectures hypergeometric-function