A few days back, I posted some #AnimatedGifs of the exact solution for a large-amplitude undamped, unforced #Pendulum. I then thought to complete the study to include the case when it has been fed enough #energy to allow it just to undergo #FullRotations, rather than just #oscillations. Well, it turns out that it is “a bit more complicated than I first expected” but I finally managed it.
#Mathematics #AppliedMathematics #SpecialFunctions #DynamicalSystems #NonlinearPhenomena
from "Definite integration using the generalized hypergeometric functions" by Ioannis Dimitrios Avgoustis (1977)
https://dspace.mit.edu/handle/1721.1/16269?utm_source=dlvr.it&utm_medium=mastodon
Power series for the cotangent and cosecant functions can be expressed rather compactly in terms of the Riemann zeta and Dirichlet eta functions:
\[ \displaystyle \cot z = -2 \sum_{k=0}^\infty \frac{ \zeta(2k) }{ \pi^{2k} } z^{2k-1} \hspace{5em}
\csc z = 2 \sum_{k=0}^\infty \frac{ \eta(2k) }{ \pi^{2k} } z^{2k-1} \]
Since I haven't seen these expressed quite this way before, I thought I'd share it. More information is available here:
https://analyticphysics.com/Special%20Functions/Zeta%20Functions%20in%20Trigonometry.htm
from "On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)" by R.W. Batterman (2007)
http://philsci-archive.pitt.edu/2629/?utm_source=dlvr.it&utm_medium=mastodon
from "q-Stirling numbers: A new view" by Yue Cai and Margaret A. Readdy (2017)
from "Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter" by Sun Yi, Patrick W. Nelson, and A. Galip Ulsoy (2007)
https://pubmed.ncbi.nlm.nih.gov/17658931/?utm_source=dlvr.it&utm_medium=mastodon
#math #delaydifferentialequations #specialfunctions #lambertw
In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and …
from "Polylogarithms and Associated Functions" by Leonard Lewin (1981)
Pustam | पुस्तम | পুস্তম🇳🇵
Khurram Wadee ✅
ƧƿѦςɛ♏ѦਹѤʞ
Christos Argyropoulos MD, PhD
tom
Paul Masson