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#specialfunctions

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ƧƿѦςɛ♏ѦਹѤʞ<p>Relations between special functions:<br><a href="https://www.johndcook.com/blog/special_function_diagram/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://www.</span><span class="ellipsis">johndcook.com/blog/special_fun</span><span class="invisible">ction_diagram/</span></a><br><a href="https://mastodon.social/tags/maths" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>maths</span></a> <a href="https://mastodon.social/tags/mathematics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>mathematics</span></a> <a href="https://mastodon.social/tags/functions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>functions</span></a> <a href="https://mastodon.social/tags/SpecialFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>SpecialFunctions</span></a> <a href="https://mastodon.social/tags/JohnDCook" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>JohnDCook</span></a> <a href="https://mastodon.social/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>math</span></a></p>
Christos Argyropoulos MD, PhD<p><span class="h-card" translate="no"><a href="https://chirp.social/@Perl" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@<span>Perl</span></a></span> One of the things I learned tonight is that the <a href="https://mstdn.science/tags/specialfunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>specialfunctions</span></a> and <a href="https://mstdn.science/tags/statisticaldistributions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>statisticaldistributions</span></a> library beating inside <a href="https://mstdn.science/tags/rstats" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>rstats</span></a> is available as a standalone version. While this speaks volumes of the portability of <a href="https://mstdn.science/tags/clang" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>clang</span></a>, 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 <a href="https://mstdn.science/tags/perl" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>perl</span></a> <a href="https://mstdn.science/tags/pdl" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>pdl</span></a> project</p>
tom<p>from &quot;Definite integration using the generalized hypergeometric functions&quot; by Ioannis Dimitrios Avgoustis (1977)</p><p><a href="https://dspace.mit.edu/handle/1721.1/16269?utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">dspace.mit.edu/handle/1721.1/1</span><span class="invisible">6269?utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a></p>
Paul Masson<p>Power series for the cotangent and cosecant functions can be expressed rather compactly in terms of the Riemann zeta and Dirichlet eta functions:</p><p>\[ \displaystyle \cot z = -2 \sum_{k=0}^\infty \frac{ \zeta(2k) }{ \pi^{2k} } z^{2k-1} \hspace{5em} <br />\csc z = 2 \sum_{k=0}^\infty \frac{ \eta(2k) }{ \pi^{2k} } z^{2k-1} \] </p><p>Since I haven&#39;t seen these expressed quite this way before, I thought I&#39;d share it. More information is available here: </p><p><a href="https://analyticphysics.com/Special%20Functions/Zeta%20Functions%20in%20Trigonometry.htm" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">analyticphysics.com/Special%20</span><span class="invisible">Functions/Zeta%20Functions%20in%20Trigonometry.htm</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a> <a href="https://mathstodon.xyz/tags/trig" class="mention hashtag" rel="tag">#<span>trig</span></a> <a href="https://mathstodon.xyz/tags/trigonometry" class="mention hashtag" rel="tag">#<span>trigonometry</span></a> <a href="https://mathstodon.xyz/tags/zeta" class="mention hashtag" rel="tag">#<span>zeta</span></a> <a href="https://mathstodon.xyz/tags/eta" class="mention hashtag" rel="tag">#<span>eta</span></a></p>
tom<p>from &quot;On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)&quot; by R.W. Batterman (2007)</p><p><a href="http://philsci-archive.pitt.edu/2629/?utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">http://</span><span class="ellipsis">philsci-archive.pitt.edu/2629/</span><span class="invisible">?utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a></p>
tom<p>from &quot;q-Stirling numbers: A new view&quot; by Yue Cai and Margaret A. Readdy (2017)</p><p><a href="https://www.sciencedirect.com/science/article/pii/S019688581630121X?utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">sciencedirect.com/science/arti</span><span class="invisible">cle/pii/S019688581630121X?utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a> <a href="https://mathstodon.xyz/tags/qcalculus" class="mention hashtag" rel="tag">#<span>qcalculus</span></a> <a href="https://mathstodon.xyz/tags/stirling" class="mention hashtag" rel="tag">#<span>stirling</span></a></p>
tom<p>from &quot;Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter&quot; by Sun Yi, Patrick W. Nelson, and A. Galip Ulsoy (2007)</p><p><a href="https://pubmed.ncbi.nlm.nih.gov/17658931/?utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">pubmed.ncbi.nlm.nih.gov/176589</span><span class="invisible">31/?utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/delaydifferentialequations" class="mention hashtag" rel="tag">#<span>delaydifferentialequations</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a> <a href="https://mathstodon.xyz/tags/lambertw" class="mention hashtag" rel="tag">#<span>lambertw</span></a></p>
tom<p>from &quot;Polylogarithms and Associated Functions&quot; by Leonard Lewin (1981)</p><p><a href="https://www.google.com/books/edition/Polylogarithms_and_Associated_Functions/yETvAAAAMAAJ?hl=en&amp;utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">google.com/books/edition/Polyl</span><span class="invisible">ogarithms_and_Associated_Functions/yETvAAAAMAAJ?hl=en&amp;utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a> <a href="https://mathstodon.xyz/tags/polylogarithm" class="mention hashtag" rel="tag">#<span>polylogarithm</span></a></p>
tom<p>from &quot;Special Functions of Mathematical Physics and Chemistry&quot; by Ian N Sneddon (1956)</p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a> <a href="https://mathstodon.xyz/tags/physics" class="mention hashtag" rel="tag">#<span>physics</span></a> <a href="https://mathstodon.xyz/tags/chemistry" class="mention hashtag" rel="tag">#<span>chemistry</span></a></p>
tom<p>from &quot;Generalized Hypergeometric Functions&quot; by Bernard Dwork (1990)</p><p><a href="https://global.oup.com/academic/product/generalized-hypergeometric-functions-9780198535676?lang=en&amp;cc=au&amp;utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">global.oup.com/academic/produc</span><span class="invisible">t/generalized-hypergeometric-functions-9780198535676?lang=en&amp;cc=au&amp;utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a></p>
Paul Masson<p>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 \( 0 &lt; \operatorname{Re} z &lt; 1 \), the restriction to the critical line is spooky cool.</p><p>With an implementation of the zeta function in <a href="https://mathstodon.xyz/tags/JavaScript" class="mention hashtag" rel="tag">#<span>JavaScript</span></a> one has a proof near the origin via <a href="https://mathstodon.xyz/tags/visualization" class="mention hashtag" rel="tag">#<span>visualization</span></a>. The real part of the function is blue, imaginary red:</p><p><a href="https://mathcell.org/www/riemann-zeta-zeros.htm" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathcell.org/www/riemann-zeta-</span><span class="invisible">zeros.htm</span></a></p><p>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</p><p>±14.13, ±21.02, ±25.01, ±30.42, ±32.94, ±37.59, ±40.92, ±43.33, ±48.01, ±49.77</p><p>For more context and the relation to the Riemann xi function, visit</p><p><a href="https://analyticphysics.com/Special%20Functions/Visualizing%20Riemann%20Zeta%20Function%20Zeros.htm" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">analyticphysics.com/Special%20</span><span class="invisible">Functions/Visualizing%20Riemann%20Zeta%20Function%20Zeros.htm</span></a></p><p><a href="https://mathstodon.xyz/tags/SpecialFunctions" class="mention hashtag" rel="tag">#<span>SpecialFunctions</span></a> <a href="https://mathstodon.xyz/tags/Riemann" class="mention hashtag" rel="tag">#<span>Riemann</span></a> <a href="https://mathstodon.xyz/tags/zeta" class="mention hashtag" rel="tag">#<span>zeta</span></a></p>
Paul Masson<p><span class="h-card" translate="no"><a href="https://mathstodon.xyz/@tomcuchta" class="u-url mention">@<span>tomcuchta</span></a></span> how about 3D versions of the direct functions?</p><p><a href="https://paulmasson.github.io/math/docs/functions/airyAi.html" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">paulmasson.github.io/math/docs</span><span class="invisible">/functions/airyAi.html</span></a><br /><a href="https://paulmasson.github.io/math/docs/functions/airyBi.html" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">paulmasson.github.io/math/docs</span><span class="invisible">/functions/airyBi.html</span></a></p><p><a href="https://mathstodon.xyz/tags/SpecialFunctions" class="mention hashtag" rel="tag">#<span>SpecialFunctions</span></a></p>
tom<p>from &quot;Airy functions and applications to physics&quot; by Olivier Vallee and Manuel Soares (2010)</p><p><a href="https://www.amazon.com/Airy-Functions-Applications-Physics-2nd/dp/184816548X?utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://www.</span><span class="ellipsis">amazon.com/Airy-Functions-Appl</span><span class="invisible">ications-Physics-2nd/dp/184816548X?utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a></p>
tom<p>from &quot;q-Special functions, a tutorial&quot; by Tom Koornwinder (2013)</p><p><a href="https://arxiv.org/abs/math/9403216?utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">arxiv.org/abs/math/9403216?utm</span><span class="invisible">_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a></p>
tom<p>From &quot;The analytic continuation of the Gaussian hypergeometric function 2F1(a,b;c;z) for arbitrary parameters&quot; by W. Becken and P. Schmelcher (2000)</p><p><a href="https://core.ac.uk/download/pdf/82108003.pdf?utm_source=dlvr.it&amp;utm_medium=mastodon" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">core.ac.uk/download/pdf/821080</span><span class="invisible">03.pdf?utm_source=dlvr.it&amp;utm_medium=mastodon</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="tag">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a></p>
Paul Masson<p>If you want to do <a href="https://mathstodon.xyz/tags/QuantumMechanics" class="mention hashtag" rel="tag">#<span>QuantumMechanics</span></a> in <a href="https://mathstodon.xyz/tags/HigherDimensions" class="mention hashtag" rel="tag">#<span>HigherDimensions</span></a> 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:</p><p><a href="https://analyticphysics.com/Special%20Functions/Hypergeometric%20Orthogonal%20Polynomials.htm" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">analyticphysics.com/Special%20</span><span class="invisible">Functions/Hypergeometric%20Orthogonal%20Polynomials.htm</span></a></p><p>Lots of tedious detail that ultimately simplifies nicely. Suspect I&#39;m missing something important here...</p><p><a href="https://mathstodon.xyz/tags/Physics" class="mention hashtag" rel="tag">#<span>Physics</span></a> <a href="https://mathstodon.xyz/tags/SpecialFunctions" class="mention hashtag" rel="tag">#<span>SpecialFunctions</span></a></p>
Nicolas Tessore<p>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? <a href="https://mathstodon.xyz/tags/numerics" class="mention hashtag" rel="tag">#<span>numerics</span></a> <a href="https://mathstodon.xyz/tags/numericalmethods" class="mention hashtag" rel="tag">#<span>numericalmethods</span></a> <a href="https://mathstodon.xyz/tags/specialfunctions" class="mention hashtag" rel="tag">#<span>specialfunctions</span></a></p>
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