Re: what actually is in the WRI "functions" database?

*To*: mathgroup at smc.vnet.net*Subject*: [mg48608] Re: what actually is in the WRI "functions" database?*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Mon, 7 Jun 2004 05:33:44 -0400 (EDT)*Organization*: The University of Western Australia*References*: <c9tn1f$sf0$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <c9tn1f$sf0$1 at smc.vnet.net>, Richard Fateman <rfateman at sbcglobal.net> wrote: > I was browsing through the WRI function database, actually > to see what indexing method was being used. But then I began > to wonder how some of the specific formulas fit into Mathematica. I tried > (the first) equation I picked on in Mathematica 5.0. > > It was formula > http://functions.wolfram.com/01.09.23.0002.01 > > which has a condition that n is a positive integer. > > This is displayed on the functions web site as n \[element] > ?[DoubleStruckCapitalN]^{+} where I've made up some of > the notation there, using TeX notation. Mathematica has a superscriptbox > notation, I think.. > > The InputForm on the functions web site says to type this into Mathemaitca as > > n \[Element] Integers && n > 0 > > which is not the same. They are, of course, equivalent. Mathematica does not have the (positive) Natural numbers as a built-in domain. > Then I looked further, nearby.. > > http://functions.wolfram.com/ElementaryFunctions/Cot/23/01/0005/ > > where there is a formula containing an ellipsis ... > > and the InputForm basically is not computationally equivalent > at all to the semantics of the formula. It just has an ellipsis! > > To summarize: > 1. There is a typeset formula T, using typical math notation. > > 2. There is an InputForm, S which is not the same as T, and probably > cannot be automatically mapped onto T from Mathematica. Actually, I think that this can be done -- see below. > 3. S, in general, does have the semantics of T either. > > 4. (oh, also), There is a MathML form. It seems to have a typeset > component that looks like T, but very verbose, and a MathML content > that is (I guess) supposed to translate into S. > In the example http://functions.wolfram.com/01.09.23.0002.01 > it is NOT the same as S, at least if you believe there is > a difference between the integers and the POSITIVE integers. I must still be missing your point here: The (set of) positive natural numbers is identical to the positive integers. > Question: Has anyone (else) found this troublesome? It is troublesome -- but the site is still extremely useful. Even if I still have to some "translation" it is a lot less than that required when reading most mathematical handbooks. As a particular example that arose in my research recently, compare Abramowitz and Stegun 16.23.10 (which, incidentally is incorrect in the edition I possess) to http://functions.wolfram.com/EllipticFunctions/JacobiNS/06/02/ The form at the functions site is immediately more useful in that the fact q depends on m is made explicit, as is the dependency of the argument of the sin function on K(m). > Is there just a disconnect between the Functions web site and what (I think) > was the intention of making it meaningful to automated mathematics? There is the Notations link (in the "menu" on each page) that takes you to http://functions.wolfram.com/Notations/ where there is a Notebook in which (most of) the notations used are explained. However, I agree that ellipsis is used without explanation -- and, of course, it has a context-dependent meaning. Nevertheless, I think that it is possible to extend Mathematica input notations using the Mathematica Notation package so that S is the same as T and automatically maps onto T within Mathematica. I have addressed the two examples you presented here using this package at http://physics.uwa.edu.au/pub/Mathematica/MathGroup/FunctionNotations.nb Also, it would be useful to extend the functions website (and/or Notations.nb) to give a table of equivalent Mathematica expressions for missing mathematical notations. For example, the set of natural numbers (Element[n,Integers] && n >= 0) and positive natural numbers (Element[n,Integers] && n > 0). > The idea that a table or encyclopedia of computerized mathematics > should be a collection of typeset math and an inaccurate rendition > of it in some computer algebra system is not particularly attractive. I think that you are being overly critical. If you can point to a better and more useful site I would love to hear about it! Cheers, Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul