Newbetuts

Integral $\int_0^\infty F(x)\,F\left(x\,\sqrt2\right)\frac{e^{-x^2}}{x^2} \, dx$ involving Dawson's function

Calculate $\int_{0}^\infty\frac{dx}{\left(1+\frac{x^3}{1^3}\right)\left(1+\frac{x^3}{2^3}\right)\left(1+\frac{x^3}{3^3}\right)\ldots}$

Strange symmetry regarding sum $\sum_{n=0}^\infty\frac{n^ne^{-bn}}{\Gamma(n+1)}$ and integral $\int_{0}^\infty\frac{x^xe^{-bx}}{\Gamma(x+1)}dx$

Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$

The Gamma function and the Pi function

Re-Expressing the Digamma

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Calculating in closed form $\int _0^1\int _0^1\frac{1}{1+x y (x+y)} \ dx \ dy$

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

Upper bound on differences of consecutive zeta zeros

Closed form for $_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right)$

The trigonometric solution to the solvable DeMoivre quintic?

Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$

Definite integral of arcsine over square-root of quadratic

Closed-form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

Strange closed forms for hypergeometric functions

An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$

New posts in special-functions

Integral $\int_0^\infty F(x)\,F\left(x\,\sqrt2\right)\frac{e^{-x^2}}{x^2} \, dx$ involving Dawson's function

doubly periodic functions as tessellations (other than parallelograms)

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

How does one calculate the amount of time required for computation?

Calculate $\int_{0}^\infty\frac{dx}{\left(1+\frac{x^3}{1^3}\right)\left(1+\frac{x^3}{2^3}\right)\left(1+\frac{x^3}{3^3}\right)\ldots}$

Strange symmetry regarding sum $\sum_{n=0}^\infty\frac{n^ne^{-bn}}{\Gamma(n+1)}$ and integral $\int_{0}^\infty\frac{x^xe^{-bx}}{\Gamma(x+1)}dx$

Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$

The Gamma function and the Pi function

Re-Expressing the Digamma

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Calculating in closed form $\int _0^1\int _0^1\frac{1}{1+x y (x+y)} \ dx \ dy$

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

Upper bound on differences of consecutive zeta zeros

Closed form for $_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right)$

The trigonometric solution to the solvable DeMoivre quintic?

Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$

Definite integral of arcsine over square-root of quadratic

Closed-form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

Strange closed forms for hypergeometric functions

An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$