#specialfunctions — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #specialfunctions, aggregated by home.social.
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@AdrianRiskin
Sketching the function in my head it's pretty clear that it has a maximum somewhere between -2 and 0. Doing the algebra I get x=exp(W(3e-1))-3 = −1.1454... which makes me confident :-)
Nice problem - wrong course.
#maths #SpecialFunctions #LambertW -
In mathematics, theta functions are special functions of several complex variables. They appear in various topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.
#ThetaFunction #JacobiThetaFunction #SpecialFunctions #ComplexVariables -
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
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Relations between special functions:
https://www.johndcook.com/blog/special_function_diagram/
#maths #mathematics #functions #SpecialFunctions #JohnDCook #math -
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
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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
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from "q-Stirling numbers: A new view" by Yue Cai and Margaret A. Readdy (2017)
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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
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from "Polylogarithms and Associated Functions" by Leonard Lewin (1981)
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from "Special Functions of Mathematical Physics and Chemistry" by Ian N Sneddon (1956)
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from "Generalized Hypergeometric Functions" by Bernard Dwork (1990)
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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 \( 0 < \operatorname{Im} z < 1 \), 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 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
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@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 -
from "Airy functions and applications to physics" by Olivier Vallee and Manuel Soares (2010)
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from "q-Special functions, a tutorial" by Tom Koornwinder (2013)
https://arxiv.org/abs/math/9403216?utm_source=dlvr.it&utm_medium=mastodon
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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
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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...
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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