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MathGroup Archive 2004

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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48621] Re: what actually is in the WRI "functions" database?
  • From: Richard Fateman <rfateman at sbcglobal.net>
  • Date: Tue, 8 Jun 2004 00:48:12 -0400 (EDT)
  • References: <c9tn1f$sf0$1@smc.vnet.net> <paul-571C5D.13304507062004@news.uwa.edu.au>
  • Sender: owner-wri-mathgroup at wolfram.com

Paul Abbott wrote:

>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.
>
If you allow equivalence, then there is no need for Mathematica's 
Simplify function.

Are you saying that the human version is inadequate and humans should 
learn to be
clear like Mathematica and say what they need to say using a simple set of
operations, and not proliferate redundant notations? That is the best spin
I can put on it, and it has limited appeal for the purposes of the functions
web site.

I suggest you try to get this NON equivalent formula the way I did:

Sum[Cot[(Pi*(2*k - 1))/(4*n)]^2, {k, 1, n}] == 
   2*n^2 - n /; n ? Integers

from
taking
http://functions.wolfram.com/01.09.23.0002.01
click on the MathML, and then request
rendering the MathML into InputForm.

I believe that  n ? Integers   includes n==0.  Do you disagree?

So the functions database has an error.  Considering this was
the first, randomly chosen, formula that I picked out, I found
this troublesome.  (To be honest, I had mixed feelings, since
picking up on faults gives me some satisfaction, too.)


....<snip>

>
>  
>
>>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). 
>

How do you know that the the formula in the WRI database is right and 
A&S is wrong?
How do you think others deal with the "troublesome" issue?  What I think 
I would do
is not trust any formula at all, and waste a lot of time checking, if I 
could, or maybe
just not use it at all. Or use Gradshteyn & Rhyzik.

>
>
>>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
>

Thanks for putting together this response to show how relatively simple 
it would be to fix this
particular fault.  I hope Michael Trott makes use of this, and also 
makes use of other ellipsis substitutions
 before he puts a few hundred more ellipsis formulas in there.

I found the formula in question, even given in standard form to be 
confusing, by the way., 

is  f(l_0) .... f(l_{2m})     something like  f(1_0), f(l_1), f(l_2) 
....   or does it include only the EVEN
entries?   

Also, the bracketing of the sigmas is confusing.  Since the big Product 
at the end depends only on l_0,
why is it (apparently) inside all the sigmas?  The input form makes the 
brackets clearer.



<snip>

>>Fateman: 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.
>>
>
>Abbott: 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!
>
You think that this effort is above criticism?  I am pointing out that I 
observe
 the website to be a repository of (perhaps) accurate typeset formulas
accompanied by non-equivalent statements in MathML and InputForm.  I think
that is a serious problem.

Are you familiar with the ESF project?  The last I looked at it, it was 
not as comprehensive, but it is (so far as I can
tell) likely to have only correct formulas.

Then there is DLMF from NIST. I have been disappointed with the 
relationship with CASs
but I doubt that anyone associated with that project would  tolerate 
statements that were only "usually" correct.

RJF


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