Lmst

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

Relations between special functions:
https://www.johndcook.com/blog/special_function_diagram/
#maths #mathematics #functions #SpecialFunctions #JohnDCook #math

@Perl One of the things I learned tonight is that the #specialfunctions and #statisticaldistributions library beating inside #rstats is available as a standalone version. While this speaks volumes of the portability of #clang, it also creates opportunities for transporting a significant chunk of R's functionalities into other languages, e.g. by writing swig interfaces. This may be an interesting #perl #pdl project

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

#math #specialfunctions

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

#math #specialfunctions #trig #trigonometry #zeta #eta

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

#math #specialfunctions

from "q-Stirling numbers: A new view" by Yue Cai and Margaret A. Readdy (2017)

https://www.sciencedirect.com/science/article/pii/S019688581630121X?utm_source=dlvr.it&utm_medium=mastodon

#math #specialfunctions #qcalculus #stirling

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

from "Polylogarithms and Associated Functions" by Leonard Lewin (1981)

https://www.google.com/books/edition/Polylogarithms_and_Associated_Functions/yETvAAAAMAAJ?hl=en&utm_source=dlvr.it&utm_medium=mastodon

#math #specialfunctions #polylogarithm

from "Special Functions of Mathematical Physics and Chemistry" by Ian N Sneddon (1956)

#math #specialfunctions #physics #chemistry

from "Generalized Hypergeometric Functions" by Bernard Dwork (1990)

https://global.oup.com/academic/product/generalized-hypergeometric-functions-9780198535676?lang=en&cc=au&utm_source=dlvr.it&utm_medium=mastodon

#math #specialfunctions

The unproved Riemann hypothesis states that the nontrivial zeros of the Riemann zeta function occur only on the critical line \( z = \frac12 + i y \). While it is not difficult to understand why these zeros can only occur inside the critical strip \( 0 < \operatorname{Re} z < 1 \), the restriction to the critical line is spooky cool.

With an implementation of the zeta function in #JavaScript one has a proof near the origin via #visualization. The real part of the function is blue, imaginary red:

https://mathcell.org/www/riemann-zeta-zeros.htm

Manipulating the imaginary part of the argument along the critical strip shows immediately that zeros only occur on the critical line for an imaginary part of approximately

±14.13, ±21.02, ±25.01, ±30.42, ±32.94, ±37.59, ±40.92, ±43.33, ±48.01, ±49.77

For more context and the relation to the Riemann xi function, visit

https://analyticphysics.com/Special%20Functions/Visualizing%20Riemann%20Zeta%20Function%20Zeros.htm

#SpecialFunctions #Riemann #zeta

@tomcuchta how about 3D versions of the direct functions?

https://paulmasson.github.io/math/docs/functions/airyAi.html
https://paulmasson.github.io/math/docs/functions/airyBi.html

#SpecialFunctions

from "Airy functions and applications to physics" by Olivier Vallee and Manuel Soares (2010)

https://www.amazon.com/Airy-Functions-Applications-Physics-2nd/dp/184816548X?utm_source=dlvr.it&utm_medium=mastodon

#math #specialfunctions

from "q-Special functions, a tutorial" by Tom Koornwinder (2013)

https://arxiv.org/abs/math/9403216?utm_source=dlvr.it&utm_medium=mastodon

#math #specialfunctions

From "The analytic continuation of the Gaussian hypergeometric function 2F1(a,b;c;z) for arbitrary parameters" by W. Becken and P. Schmelcher (2000)

https://core.ac.uk/download/pdf/82108003.pdf?utm_source=dlvr.it&utm_medium=mastodon

#math #specialfunctions

If you want to do #QuantumMechanics in #HigherDimensions then you need to know about associated Gegenbauer polynomials. Since there is no good reference on the web for these, I put together a presentation of Legendre, Gegenbauer and Jacobi polynomials to show how to derive their series expansions, Rodrigues formulas and differential equations:

https://analyticphysics.com/Special%20Functions/Hypergeometric%20Orthogonal%20Polynomials.htm

Lots of tedious detail that ultimately simplifies nicely. Suspect I'm missing something important here...

#Physics #SpecialFunctions

Any special function specialists around here? Are there good numerical methods to evaluate bivariate (hyper2d) hypergeometric functions/Kampé de Fériet functions besides evaluating the inner hypergeometric function and summing up? #numerics #numericalmethods #specialfunctions