New posts in special-functions

Interesting integral related to the Omega Constant/Lambert W Function

definite-integrals special-functions lambert-w

Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?

calculus special-functions closed-form gamma-function

Calculating alternating Euler sums of odd powers

integration sequences-and-series definite-integrals special-functions euler-sums

Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

integration special-functions gamma-function riemann-zeta zeta-functions

Roots of the incomplete gamma function

special-functions roots gamma-function

Closed-form for $\sum_{n=1}^{\infty} \left(m n \, \text{arccoth} \, (m n) - 1\right)$

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

Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$

calculus special-functions closed-form conjectures hypergeometric-function

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?

analysis special-functions gamma-function

Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

special-functions improper-integrals closed-form

Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

probability special-functions binomial-coefficients definite-integrals

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc... & Representation Theory of Special Functions

ordinary-differential-equations representation-theory special-functions lie-groups

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

calculus integration special-functions definite-integrals closed-form

Solving Special Function Equations Using Lie Symmetries

ordinary-differential-equations representation-theory special-functions

What are BesselJ functions?

real-analysis integration analysis special-functions bessel-functions

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

calculus sequences-and-series trigonometry special-functions

Identification of a curious function

calculus sequences-and-series functions special-functions game-theory

Why is Euler's Gamma function the "best" extension of the factorial function to the reals?

real-analysis education gamma-function special-functions

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

integration definite-integrals special-functions

How do you prove Gautschi's inequality for the gamma function?

inequality special-functions gamma-function