New posts in special-functions

Looking for a proof of Cleo's result for ${\large\int}_0^\infty\operatorname{Ei}^4(-x)\,dx$

calculus integration definite-integrals special-functions closed-form

Prove ${\large\int}_{-1}^1\frac{dx}{\sqrt[3]{9+4\sqrt5\,x}\ \left(1-x^2\right)^{2/3}}=\frac{3^{3/2}}{2^{4/3}5^{5/6}\pi }\Gamma^3\left(\frac13\right)$

calculus definite-integrals special-functions closed-form hypergeometric-function

How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?

special-functions gamma-function closed-form conjectures hypergeometric-function

Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?

sequences-and-series integration special-functions analytic-number-theory

Beta function derivation

probability-theory statistics special-functions

Polygamma function series: $\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

calculus integration sequences-and-series inequality special-functions

Is this function decreasing on $(0,1)$?

real-analysis reference-request special-functions elliptic-integrals

Short calculation of the dilogarithm?

special-functions polylogarithm

What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

special-functions binomial-coefficients analytic-number-theory riemann-zeta zeta-functions

Product $\left(\frac31\right)^{1/2}\cdot\left(\frac75\right)^{1/6}\cdot \left(\frac{11}9\right)^{1/10}\cdot \left(\frac{15}{13}\right)^{1/14}\cdots$

calculus real-analysis integration sequences-and-series special-functions

Is this series known? [closed]

real-analysis calculus sequences-and-series analysis special-functions

Fibonorial of a fractional or complex argument

sequences-and-series recurrence-relations special-functions fibonacci-numbers analyticity

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

calculus integration special-functions closed-form hypergeometric-function

Real and imaginary part of Gamma function

special-functions gamma-function riemann-zeta

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

analysis integration special-functions improper-integrals

When is an elliptic integral expressible in terms of elementary functions?

calculus integration special-functions

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,\mathrm{d}x$ have a closed form?

integration definite-integrals special-functions closed-form

About the Legendre differential equation

sequences-and-series ordinary-differential-equations special-functions

Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$

sequences-and-series special-functions gamma-function closed-form combinatorics

Integrate $\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$

calculus trigonometry special-functions definite-integrals closed-form