Let $\varpi ={\displaystyle \frac{{\mathrm{\Gamma }}^2\left(\frac{1}{4}\right)}{2\sqrt{2\pi }}}=2.62205755\dots$ be the lemniscate constant. Then,
$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{1}{{\sinh }^4(\pi n)}}={\displaystyle \frac{{\varpi }^4}{30{\pi }^4}}+{\displaystyle \frac{1}{3\pi }}-{\displaystyle \frac{11}{90}}}$$

A lattice sum with a surprising closed form:
$${\displaystyle \sum _{\begin{array}{c}(m,n)\in {\mathbb{Z}}^2\\ (m,n)\ne 0\end{array}}{\displaystyle \frac{1}{(m+ni{)}^4}}={\displaystyle \frac{{\varpi }^4}{15}}={\displaystyle \frac{{\mathrm{\Gamma }}^8\left(\frac{1}{4}\right)}{960{\pi }^2}}}$$
#LatticeSum #ClosedForm #LemniscateConstant #Series #Sum #GammaFunction #Pi

A small project from yesterday, plotting Gamma(x) = Gamma(y). Wolfram Alpha (left) is awful. Desmos is OK but not great. My custom code fills in the gaps, but is that really what a graph should show, or should it highlight all the discontinuities?

How to prove this?!

$${\displaystyle {\int }_{\frac{1}{2}}^1{\displaystyle \frac{\psi (x)}{1+{\mathrm{\Gamma }}^2(x)}}dx={\displaystyle \frac{1}{2}}\ln \left({\displaystyle \frac{1+\pi }{2\pi }}\right)}$$

where $\mathrm{\Gamma }(x)$ and $\psi (x)$ are the gamma and digamma functions respectively.

The Riemann hypothesis, explained by Jørgen Veisdal: https://www.cantorsparadise.com/the-riemann-hypothesis-explained-fa01c1f75d3f
$$\boxed{{\displaystyle \zeta (s)=0,s\notin 2{\mathbb{Z}}^{-}⟹\mathrm{\Re }(s)={\displaystyle \frac{1}{2}}}}$$

GAUTSCHI'S INEQUALITY:

$\bullet$ Let $x\in {\mathbb{R}}^{+}$ and $s\in (0,1)$. Then, an inequality for ratios of gamma functions known as Gautschi's inequality:
$$x^{1-s}<{\displaystyle \frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x+s)}}<(x+1{)}^{1-s}$$
$\bullet$ Asymptotic behaviour of the ratios of gamma functions:
$${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x+s)x^{1-s}}}=1}$$
#GammaFunction #GautaschiInequality #Inequality #RealAnalysis #AsymptoticBehaviour #mathematics #analysis #lowerbound #upperbound