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

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
DLMF
<http://dlmf.nist.gov/Contents/TJ/synopsis.php>
but did not receive a response.

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
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

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