## SFC - Special Function Concordance

**The goal of SFC is the formulation and communication of semantic information in a way that promotes interoperability of math.**

The use of Special Functions is among the less elementary subjects that are significant in engineering and other applications of mathematics. We envisage a service that will allow the use of special functions to be even more reliable. The service will also clarify the relationships between the capabilities provided by the many existing tools.

The mathematics of a great number of special functions is so well understood that computer algebra systems, both commercial and non-commercial, offer calculation with such functions, both numerical and symbolical.

The implementations of trigonometric functions, Bessel, Legendre, Chebyshev, functions in Maple, Mathematica, Matlab, SageMath, Pari, Gap and other such systems have, to a large extent, replaced the old uses of tables and handbooks in dealing with these mathematical objects.

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The classic Handbook by Abramowitz and Stegun from the National Bureau of Standards has been put out in a revised edition, with an online form as the Digital Library of Mathematical Functions from NIST (National Institute of Standards and Technology). On the Web, this joined the massive online Wolfram Functions site, publicly available due to support from the commercial entity Wolfram Research, producers of Mathematica. There are also offerings from the creators of Maple, a Digital Dictionary of Mathematical Functions from INRIA in France, and other repositories of information about special functions.

This, however, does not mean that there is healthy competitive effort or that it does not matter which tools are being used. One tool might be more efficient for a particular calculation, but the mathematical facts should be the same everywhere. In truth, this is not so. There are almost as many different views of special function as there are systems treating them. They often do so without an unwitting user’s being aware of it.

There has been a sense in which it has been natural for each system—Mathematica (now the Wolfram Language), Maple, Matlab, NAG, etc. —to stick to and promote its own conventions, with little interest in comparison, or even in interoperability, with others. This is, of course, unsatisfactory in order to build the sorts of tools there should be for technology and the development of science.

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The GDML WG has been aware of these problems since its inception. A GDML should be able to work with mathematical knowledge in equivalent representations and to know when what was thought to be the same turns out not to be.

Two members of the WG are now on the NIST board overseeing the DLMF, and several European special function efforts have also had contact with the WG. In addition, there has been concern about special functions displayed by the OpenMath Society. The OpenMath markup is coherent at a basic level with Content markup of MathML, which was developed by the W3C (World Wide Web Consortium) and is now an ISO standard.

At the recent gatherings in Seattle (Jan 2016) and Toronto (Feb 2016), matters of concern to a GDML were worked over. Representatives of both Mathematica and Maple publicly declared their willingness to consider, in principle, working on this according their notions of specific functions. The same general support of the idea was to be heard from other parties at the OPSF13 meeting at NIST in June 2015.

NIST is suggested as a broker who is obliged by law to be neutral. One current proposal is to organize groups around NIST, working on setting down the vocabularies, definitions and conventions used to describe the well-known special functions. The groups would, in addition, record the comparisons of their various forms in such a way for tools to provide interoperability for, and integration of, the vast knowledge implicit in the various systems. We hope that an IMKT legal entity can provide a way to finesse social issues so as to get to the science and eventually provide the desired public service.