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Thanks for your kind words! You’re right — concrete examples of how special functions solve real problems really help bring the subject to life. Due to time constraints and the large volume of material, this course was designed to provide a brief introduction along with links to the reference center, where more examples of real applications are available.

I’d suggest starting with the examples in the reference center, then exploring the NIST DLMF or Olver’s Asymptotics and Special Functions. And don’t worry about rabbit holes — that’s a natural part of learning, and you can always circle back with questions.

For your information, there will be an upcoming Wolfram U course, Introduction to PDEs, in a few months, prepared by my colleagues Aram Manaselyan and Devendra Kapadia. Please feel free to join that course as well.

(1) Arben Kalziqi directed me to this Stephen Wolfram post from December 13, 2021: https://writings.stephenwolfram.com/2021/12/launching-version-13-0-of-wolfram-language-mathematica/. In it he says, “When I first started using special functions about 45 years ago, the book that was the standard reference was Abramowitz & Stegun’s 1964 Handbook of Mathematical Functions. It listed hundreds of functions, some widely used, others less so. And over the years in the development of Wolfram Language we’ve steadily been checking off more functions from Abramowitz & Stegun. And in Version 13.0 we’re finally done! All the functions in Abramowitz & Stegun are now fully computable in the Wolfram Language.”

In Lesson 01 today, A&S’s successor, the NIST Handbook of Mathematical Functions from 2010 was shown (see https://www.cambridge.org/ne/universitypress/subjects/mathematics/abstract-analysis/nist-handbook-mathematical-functions) along with its companion site in Reference 4: Digital Library of Mathematical Functions (DLMF— https://dlmf.nist.gov). What percentage of the functions in the NIST HoMF and DLMF are now implemented in the Wolfram Language?

(2) Reference 1 gives the Bateman Manuscript Project’s Higher Transcendental Functions, Vol. I-III by A. Erdelyi, et al. Included in the Bateman Project are the two volume Tables of Integral Transforms. To see precisely what is contained in these five volumes, see pages 2-22 and 23-32 of the attached files (the Word docx version is much clearer than the pdf version).

There is a successor to the Bateman Project (see pages 33-42 in the attached files): “The Askey–Bateman Project is an Encyclopedia of Special Functions in three volumes, published and to be published by Cambridge University Press”. This is from the project’s webpage https://staff.fnwi.uva.nl/t.h.koornwinder/specfun/AskeyBateman.html which was “Last modified: 13 June 2023” with no indication of when Vol. III could appear.

What percentage of the functions in the Bateman Project and/or the Askey–Bateman Project are now implemented in the Wolfram Language?

(3) Another source of much information about special functions is Gradshteyn and Ryzhik Table of Integrals, Series, and Products, Eighth Edition at: https://www.sciencedirect.com/book/9780123849335/table-of-integrals-series-and-products. There is an online Seventh Edition here. The next question is, Can Wolfram reproduce or give equivalent forms of all the integrals, series, and products in G & R? Incidentally, there is a sequel of sorts to that: Special Integrals of Gradshteyn and Ryzhik: the Proofs by Victor H. Moll (Volumes 1 & 2, 2014-2015) — https://www.taylorfrancis.com/books/mono/10.1201/b17674/special-integrals-gradshteyn-ryzhik-victor-moll.

Attachments:

Bateman Manuscri...docx

Bateman Manuscri...pdf

This is just a reminder that the Special Functions Study Group begins on Monday, September 15. Tigran has worked closely with me during the last year to create this unique introduction to the 100 most fundamental special functions in the Wolfram Language, and I strongly recommend the Study Group to everyone!