Let \(\varpi=\dfrac{\Gamma^2\left(\frac14\right)}{2\sqrt{2\pi}}=2.62205755\ldots\) be the lemniscate constant. Then,
\[\Large\displaystyle\sum_{n=1}^\infty\dfrac{1}{\sinh^4(\pi n)}=\dfrac{\varpi^4}{30\pi^4}+\dfrac{1}{3\pi}-\dfrac{11}{90}\]

#Series #Sum #InfiniteSum #LemniscateConstant #GammaFunction #Lemniscate #LemniscateOfBernoulli #Bernoulli #Math #Maths #InfiniteSeries #HyperbolicSines

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?

#GammaFunction #MathVisualization

How to prove this?!🤔

\[\displaystyle\int_\tfrac{1}{2}^1\dfrac{\psi(x)}{1+\Gamma^2(x)}dx=\dfrac{1}{2}\ln\left(\dfrac{1+\pi}{2\pi}\right)\]

where \(\Gamma(x)\) and \(\psi(x)\) are the gamma and digamma functions respectively.

#Integral #definiteintegral #gammafunction #digammafunction #pi #logarithm

The Riemann hypothesis, explained by Jørgen Veisdal: 🔗 https://www.cantorsparadise.com/the-riemann-hypothesis-explained-fa01c1f75d3f
\[\boxed{\zeta(s)=0,\ s\notin2\mathbb{Z}^-\implies\Re(s)=\dfrac{1}{2}}\]

#RiemannHypothesis #Riemann #Hypothesis #Atiyah #NumberTheory #Conjecture #AnalyticNumberTheory #Mathematics #CantorsParadise #OpenProblem #UnsolvedProblem #PrimeNumber #RiemannZetaFunction #ZetaFunction #NonTrivialZero #LogarithmicIntegral #XiFunction #PrimeNumberTheorem #PrimeNumberDistribution #PrimeCountingFunction #GammaFunction

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:
\[x^{1-s}<\dfrac{\Gamma(x+1)}{\Gamma(x+s)}<(x+1)^{1-s}\]
\(\bullet\) Asymptotic behaviour of the ratios of gamma functions:
\[\displaystyle\lim_{x\rightarrow\infty}\dfrac{\Gamma(x+1)}{\Gamma(x+s)x^{1-s}}=1\]
#GammaFunction #GautaschiInequality #Inequality #RealAnalysis #AsymptoticBehaviour #mathematics #analysis #lowerbound #upperbound