#lemniscate — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #lemniscate, aggregated by home.social.
-
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
-
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
-
"Lemniscating"
Another math-meditation, this time over lemniscate spirals, bells and an ambient sound...
Inspired by @fractalkitty 🙂
In my series of Tooll3 point generators, today I created first a lemniscate, but then extended it to be able to create lemnispirals by varying the radius over time.
#Tooll3 #realtime #particles #livingpainting #creativecoding #hlsl #gpu #shader #generative #procedural #mastoart #audioreactive #sounddesign #math #lemniscate #opensource
-
An interesting infinite product!
\[\displaystyle\prod_{n=1}^\infty\coth^{(-1)^n}\left(\dfrac{\pi n}{2}\right)=\dfrac{\sqrt\pi}{\sqrt[4]2\sqrt{\varpi}}\]
where \(\varpi=2.62205755\ldots\) is the lemniscate constant (the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of \(\pi\) for the circle). Both \(\varpi\) and \(\pi\) are proven to be transcendental.
#lemniscate #lemniscateconstant #product #infiniteproduct #HyperbolicFunction #HyperbolicCotangent #Function #pi #maths #math
-
An interesting infinite product!
\[\displaystyle\prod_{n=1}^\infty\coth^{(-1)^n}\left(\dfrac{\pi n}{2}\right)=\dfrac{\sqrt\pi}{\sqrt[4]2\sqrt{\varpi}}\]
where \(\varpi=2.62205755\ldots\) is the lemniscate constant (the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of \(\pi\) for the circle). Both \(\varpi\) and \(\pi\) are proven to be transcendental.
#lemniscate #lemniscateconstant #product #infiniteproduct #HyperbolicFunction #HyperbolicCotangent #Function #pi #maths #math
-
An interesting infinite product!
\[\displaystyle\prod_{n=1}^\infty\coth^{(-1)^n}\left(\dfrac{\pi n}{2}\right)=\dfrac{\sqrt\pi}{\sqrt[4]2\sqrt{\varpi}}\]
where \(\varpi=2.62205755\ldots\) is the lemniscate constant (the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of \(\pi\) for the circle). Both \(\varpi\) and \(\pi\) are proven to be transcendental.
#lemniscate #lemniscateconstant #product #infiniteproduct #HyperbolicFunction #HyperbolicCotangent #Function #pi #maths #math
-
An interesting infinite product!
\[\displaystyle\prod_{n=1}^\infty\coth^{(-1)^n}\left(\dfrac{\pi n}{2}\right)=\dfrac{\sqrt\pi}{\sqrt[4]2\sqrt{\varpi}}\]
where \(\varpi=2.62205755\ldots\) is the lemniscate constant (the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of \(\pi\) for the circle). Both \(\varpi\) and \(\pi\) are proven to be transcendental.
#lemniscate #lemniscateconstant #product #infiniteproduct #HyperbolicFunction #HyperbolicCotangent #Function #pi #maths #math
-
An interesting infinite product!
\[\displaystyle\prod_{n=1}^\infty\coth^{(-1)^n}\left(\dfrac{\pi n}{2}\right)=\dfrac{\sqrt\pi}{\sqrt[4]2\sqrt{\varpi}}\]
where \(\varpi=2.62205755\ldots\) is the lemniscate constant (the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of \(\pi\) for the circle). Both \(\varpi\) and \(\pi\) are proven to be transcendental.
#lemniscate #lemniscateconstant #product #infiniteproduct #HyperbolicFunction #HyperbolicCotangent #Function #pi #maths #math
-
History of the Picard scheme by Steven L Kleiman
#lemniscate
#Bernoulli
#Fagnano
#Euler -
Mathober Day 31 - Unity (Last day!!! - posted early)
from gas to solid
and back again to the stars
you are – lemniscate
#mathober2022 #mathart #haiku
#math #lemniscate #unity #mindfulness #poetry