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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
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