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[WSG25] Daily Study Group: Introduction to Special Functions

A Wolfram U Daily Study Group on “Introduction to Special Functions” begins on Monday, September 15, 2025.

Join a cohort of fellow mathematics enthusiasts to learn about the fundamentals of the theory of special functions.

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Our topics will include various groups of special functions (gamma and related functions, special integrals, orthogonal polynomials, Bessel and related, hypergeometric, Heun, Appell, Meijer’s G and other classes of functions), methods of working with them, as well as powerful built-in tools of the Wolfram Language applied to special functions.

The study group will be led by me and @Devendra Kapadia and you will learn a lot about special functions in the Wolfram Language!

Some prior knowledge of Wolfram Language, calculus, complex analysis and differential equations is required.

Please feel free to use this thread to collaborate and share ideas, materials and links to other resources with fellow learners.

Dates

September 15-September 26, 2025, 11am-12pm CT (4-5pm GMT)

REGISTER HERE

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I was drawn to this DSG because about 20 years ago while I was working as an experimental research engineer in the oil and gas upstream industry. I was able to obtain an analytic solution to a system of PDEs that describe the measurement of capillary pressure (Pc) in cylindrical core samples (~2,5-3.0”L x 1.5”D) on a centrifuge using Mathematica. The core sample is mounted in a coreholder that can allow application of overburden pressure consistent with that in the subsurface. Typically, four core samples are used in each centrifuge run, The coreholders are hung on a trunyen head attached to the vertical centrifuge rotor. The coreholders are free to swing into a horizontal position once the centrifuge is started. Displaced fluid is collected in a calibrated transparent cup attached to the coreholder. The displacing fluid, also in the cup, is connected to the inlet face of core sample. The centrifuge can also operate at elevated temperatures to match that of the oil and gas reservoir of interest. The centrifuge is run at each angular velocity until the volume of fluid displaced from the core samples are constant. The centrifuge is then run at the next angular velocity. The simplest Pc measurement is to begin with the core sample pore volume filled completely with brine (to match the composition of that in the reservoir if possible). This initial condition is referred to as 100% water saturation; Sw = 1. This corresponds to the state of the reservoir before it was charged with hydrocarbons, a process known as primary drainage. The brine is displaced from the core sample with nitrogen. The cumulative volume produced at each centrifuge angular velocity is recorded, from which the average Sw in the core sample can be determined based on the measured porosity of the sample. One then assumes a constitutive equation that describes the capillary pressure as a function of Sw. For example, for primary drainage, Pc = a((Sw-Swir)/(1-Swir))^b + Pcth is often used, where a, b, Swir and Pcth are regression parameters. Swir, referred to as the irreducible water saturation, corresponds to infinite capillary pressure; i.e., Swir = Limit[Pc(Sw), Pc -> +Infinity]. Pcth is the threshold pressure, which corresponds to the minimum capillary pressure required before nitrogen can penetrate the inlet face of the core sample. In this primary drainage measurement, the inlet face of the core sample is closest to the centrifuge rotational axis. The boundary condition for the primary drainage Pc measurements is that Sw=1 at the outlet face of the core samples at all times; Sw(z=L) = 1, where L is the length of the core sample. Based on the geometric dimensions of the centrifuge and coreholders, Pc(z) can be easily calculated at each centrifuge angular velocity based on the density difference between the displacing and displaced fluids. The data analysis consists of minimizing the cumulative error between the measured Sw and that given by integrating Pc(z) from z=0 to z=L for each centrifuge speed, using regression to obtain the best fit of a, b, Swir, and Pcth. Until that time, this regression analysis was done numerically using cubic splines. The analytic solution contained a term containing the hypergeometric function, with which I was not familiar at the time. I feel I’m more familiar with it, and all the other special functions discussed in this DSG, now.

One thing I would like to see more are in depth examples of instances where these special functions provide solutions to the ODEs and PDEs that describe the physical phenomena of interest. I realize many phenomena were mentioned, but in areas in which I’m not particularly well versed. I suppose I can check out all the references provided in this DSG to start, but I’m afraid I’ll get stuck at some point and not know where to turn and end up spiraling down a rabbit hole. In any case, this DSG has piqued my interest in this rather esoteric branch of mathematics that are the special functions. Thanks for providing this valuable material!

POSTED BY: James Kralik
Posted 11 days ago

(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.

POSTED BY: Gerald Oberg
Posted 11 days ago

Can you please give the code used to produce the colorful logo?

POSTED BY: Gerald Oberg

At this point, we can confirm that Wolfram Language has full coverage of the functions in Abramowitz & Stegun, which was an important milestone in Version 13.0. For other references—the NIST Handbook/DLMF, the Bateman and Askey–Bateman Projects, and Gradshteyn & Ryzhik—systematic benchmarking is still in progress. Much of their material (functions, integrals, series, etc.) can already be reproduced in Wolfram Language, but full coverage has not yet been formally verified.

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!

POSTED BY: Devendra Kapadia
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