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

• To: mathgroup at smc.vnet.net
• Subject: [mg48632] Re: what actually is in the WRI "functions" database?
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Wed, 9 Jun 2004 04:16:48 -0400 (EDT)
• References: <c9tn1f$sf0$1@smc.vnet.net> <paul-571C5D.13304507062004@news.uwa.edu.au> <40C48234.1030503@sbcglobal.net>
• Sender: owner-wri-mathgroup at wolfram.com

On 7/6/04, Richard Fateman wrote:

...<snip>

>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?

I prefer the human version and, personally, make 
extensive use of the Notations package so that I 
can use "standard" 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 agree that it would be nice if everything on 
the functions web site could be directly entered 
into Mathematica and work as-is. However, I know 
that you appreciate that this is non-trivial, 
even after suitable notations have been defined. 
As a particular example of this consider

   http://functions.wolfram.com/GammaBetaErf/Gamma/06/02/0003/

The condition on the sum "defines" the recurrence formula for the coefficients:

   P[n, j] == (n - 1) (P[n - 1, j] + (n - 2) P[n - 3, j - 1]) &&
   P[0, 0] == 1 && P[n, 1] == (n - 1)! && P[n, j] == 0 /; n <= 3 j - 1

However, to be really useful, this condition 
needs to be translated into set of a 
computational rules, say

   P[0, 0] = 1;
   P[n_, 1] = (n - 1)!;
   P[n_, j_] := 0 /; n <= 3 j - 1
   P[n_, j_] == (n - 1) (P[n - 1, j] + (n - 2) P[n - 3, j - 1])

While this conversion is not difficult, it would 
be nice for the computation of asymptotic formula 
and coefficients (up to a specified order) be 
"automated".

....<snip>

>>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?

I do know that A&S 16.23.10 is wrong. If you compute

   Sum[4 Pi (-1)^k/(Exp[Pi (2k+1)] - 1), {k,0,Infinity}]

(this appeared on sci.math.num-analysis recently) 
using A&S 16.23.7 you get the right answer

   Gamma[1/4]^2 Pi^(-1/2)/2 - Pi

Using A&S 16.23.10 you can spot the sign error. 
Using the corresponding formulas in the functions 
database you get the right answer in both cases.

By the way, A&S is "on-line" at

   http://jove.prohosting.com/~skripty/intro.htm

and this formula is on page 575:

   http://jove.prohosting.com/~skripty/page_575.htm

>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.

You really trust G&R? Maybe the errata list at

   http://www.cnl.salk.edu/~ruderman/gr.html

covers all the errors, but I doubt it.

....<snip>

>>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.

It would be very useful but, time-consuming, of course.

>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?

Good question. I assumed it included all entries 
and this did give consistent answers but I agree 
that it is not clear as specified.

>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?

I'll leave that question to Michael Trott.

....<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?

Not at all. It's just that the other sites and 
references I know of suffer from similar or other 
worse defects, and are even less useful in that I 
have to manually translate everything into 
computer-readable form.

Also, I still do not understand your objection 
about the typeset formulae including the domain 
of positive natural numbers being implemented in 
the Mathematica code as positive integers. The 
incorrect conversion to MathML is another issue 
(and not a Mathematica error).

>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.

I agree that correctness is important, as is utility.

>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.

I had not seen ESF <http://algo.inria.fr/esf/> 
before. An interesting resource. It is a pity 
that there is only a static printed version (I 
see that there is a LaTeX version) with no MathML 
expressions reducing its usefulness.

I have only just had a brief look around but I 
immediately encountered some puzzling formulas. 
At 
http://algo.inria.fr/esf/function/BSJ/BSJ.html#BSJ 
the use of partial derivatives and defining x to 
be a complex variable is unusual -- but, of 
course, there is nothing wrong with this. I do 
not understand BSJ.1.2. The left-hand side 
involves derivatives with respect to x and the 
right-hand side is a constant? I think I know 
what the authors are trying to say, that is A&S 
9.1.7 or

  Normal[Series[BesselJ[n, z]/z^n, {z, 0, 1}]]

but this is certainly not clear.

Asymptotic formulas such as BSJ.2.2.1 are hard to 
understand, and much more complicated than 
necessary (making them difficult to implement) 
with the embedded Maple RootOf constructs, 
consecutive -ve signs, overall factors not 
extracted from the expansion, and "undefined" use 
of ellisis preceded by a factor of 2. Instead of 
RootOf, why not just +/- I? Mathematica gives 
{-I,I} for these two roots:

   Table[Root[Function[x, x^2 + 1], i], {i,2}]

As written BSJ.2.2.1 is not correct since the 
asymptotic formula should involve inverse powers 
of x. The correct formula is given in section 9.2 
of A&S or at 
http://functions.wolfram.com/BesselAiryStruveFunctions/BesselJ/06/02/01/.

It looks to me like these problems have resulted 
from naive conversion of Maple output into TeX.

>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.

I am well aware of DLMF, the NIST project to 
develop a replacement to Abramowitz and Stegun's 
Handbook of Mathematical Functions. I subscribed 
to the DLMF-ANNOUNCE mailing list in April 2001 
but have never received a mailing. At the same 
time I wrote to Dan Lozier and Leonard Maximon 
asking about the n-j coefficients entry in the 
DLMF 
<http://dlmf.nist.gov/Contents/TJ/synopsis.php> 
but did not receive a response.

The table of contents at 
http://dlmf.nist.gov/Contents/ still only has one 
live entry. I know that some chapters were 
completed over two years ago (e.g. the chapters 
on Lamé Functions and Spheroidal Wave Functions 
by Volkmer; A former MSc student has contributed 
Mathematica code for computing Spheroidal 
Harmonics). At http://dlmf.nist.gov/about/ it 
says that in 2004 the plan is to finalize and 
validate all chapters and Web site. I wonder how 
that is going?

As one comparative example of the usefulness of 
DMLF versus the functions website, surely you 
would agree that, e.g., the asymptotic formula

http://functions.wolfram.com/BesselAiryStruveFunctions/AiryAi/06/02/01/01/0004/

is more useful than having to combine AI.7.1 and AI.7.2 of

   http://dlmf.nist.gov/Contents/AI/AI.7_i.html

Cheers,
Paul