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

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  • Subject: [mg48677] Re: what actually is in the WRI "functions" database?
  • From: Richard Fateman <fateman at>
  • Date: Thu, 10 Jun 2004 02:43:21 -0400 (EDT)
  • References: <> <> <p06110400bcec42b09adc@[]>
  • Sender: owner-wri-mathgroup at

Paul Abbott wrote:

> On 8/6/04, Richard Fateman wrote:
>> Why should I believe Michael Trott's assertion that some 
>> standard-form formula
>> is correct? Who does he think he is, Stephen Wolfram? :)
> Michael should answer this. However, I would expect that:
> [1] the formulae are compared to other tables;
> [2] self-consistency checks are made (using Mathematica) within and 
> between formulae; 

Since the standard form and the mathematica input form can be, and 
apparently ARE different,
you cannot check consistency between the standard form and the 
computer-algebra form
this way!

> [3] feedback from people who have used the functions website (such as 
> myself) is used to identify errors.
> And at it says that
> "A wide variety of methods were used to derive and verify the 
> identities listed on this site. Many standard techniques were used to 
> calculate indefinite and definite integrals. Examples of other 
> frequently used methods include symbolic high-order series expansions 
> and matching coefficients (for instance, for deriving contiguous 
> relations for generalized hypergeometric functions and modular 
> equations for modular function), polynomialization and application of 
> elimination techniques (for instance, for deriving differential 
> equations for elliptic integrals and functions)."
>> And the Mathematica formulas that he knows to be wrong are not even 
>> marked as wrong (or inadequately encoded).
> Which formulas are you talking about? I'm not aware of obviously wrong 
> StandardForm formulae, though there are coding issues such as the use 
> of ellipsis. 

I have no particular reason to identify StandardForm errors, since I 
don't intend to copy down, using a pencil and
paper, any of these forms.  I am concerned only with the CAS form 
errors.  So far as I can tell, EVERY
formula with an ellipsis is wrong.  Probably wrong wrt InputForm as well 
as MathML.

>> This is, to me, quite unacceptable. Excuses after the fact like "this 
>> will have to do until a better encoding comes along in a future 
>> Mathematica"...
> I assume you're referring to the ellipsis example? These issues could 
> be resolved by coding appropriate notations using the current version 
> of Mathematica. 

One must deduce the pattern of each ellipsis and make the appropriate 
substitution.  This is not something that
can be done automatically. 

Given Mathematica's cavalier attitude about singularities of "generic" 
functions, I would not be surprised
if there were other errors.  e.g.  a formula  f(a,b,x)= g(a,b,x)  could 
be wrong for an infinite number of
specific pairs of parameters a,b,  and mathematica would still think it 
was correct.

There may be other issues as well.  When we attempted to convert G&R's 
TeX into a CAS language,
the largest group of problems were, I think, ellipsis, but there were 

>> ...or "MathML doesn't have the expressive capacity for this" are not 
>> adequate rationals to provide known erroneous formulas.
> I wonder if erroneous formulas of this type can be spotted by the 
> comparing the input and output of the cycle StandardForm -> MathML -> 
> StandardForm ? And whether this can be used as a MathML testbed?

This is truly pointless because, even if there was a mechanism to 
translate automatically from StandardForm to MathML
and back, all that would do is check the TYPESETTING, not the 
semantics.  If there is an automatic
conversion from MathML to StandardForm  (e.g. as in any 
MathML-supportive browser), then you could
throw away any "independent" StandardForm and replace it with the MathML 

>> I don't understand how this could even be controversial. If something 
>> is semantically meaningless, you don't just hand it to someone 
>> without mentioning this fact.
> A caveat to this effect could be added in such cases. I have no idea 
> how many formulae have such problems. 

For G&R, I would guess about a quarter of the formulas, but that is just 
a guess.

>> If Lozier and his DLMF team say a formula is correct, it could still 
>> be wrong, but at least we have some idea that it has been checked 
>> over by an official "team" of experts.
> Users of the functions website, including experts, have been 
> testing/checking the website for over two years now and many will have 
> supplied feedback. While I respect the DLMF experts, until DLMF is 
> live no external testing is possible. 

I agree.

>> Yet I have expressed my disappointment with the DLMF project to 
>> Lozier and others, not only on the time scale, but the general lack 
>> of ambition.
> I strongly agree here.
>> I have pointed out to them the possibility of having more of a "live" 
>> aspect, and in a talk at NIST (years ago) I had slides of pages from 
>> the functions.wolfram site.  If NIST were to do something like this, 
>> it would have more authority, being "vendor independent" and might be 
>> optionally attached to one (or more) CAS. Some might even be on-line, 
>> live, and free.
> I suggested to Hans Volkmer that, rather than hyperlinks to references 
> and code, it would be nice for the DLMF to be able to directly call 
> functions via, e.g., web form interface (in particular, I was thinking 
> of using webMathematica to call Peter Falloon's SpheroidalHarmonics 
> package). However, it appears that this sort of functionality was 
> never intended to be part of the DLMF, which is disappointing. 

I don't know Hans Volkmer.  The only technical contact I had with the 
team was with Bruce Miller.

>> G&R has been checked over by many people. It may still have mistakes, 
>> but there have been many eyes looking at it. For a long time. And it 
>> has cross references.
> I see that there is a more complete errata at 
> I agree that GR is useful, as is A&S, partially because both are well 
> tested, and I refer to both regularly. However, I now find that I go 
> to the functions website first because usually I want to compute 
> something and I can get the required formula very quickly.
>> And the latest version, edited by Dan Zwillinger, was generated from 
>> a macro language with an expansion into live CAS language 
>> (Mathematica, I think.)
> I was not aware of this. It is not mentioned at the book website but 
> I, and I'm sure many others, would be interested to know more about 
> such a capability. Especially since the functions website is 
> restricted, at present, to functions built-in to Mathematica.
>> While the concept, the typography, and the scope of functions.wolfram 
>> is admirable, the content seems to be untrustworthy, so what's the 
>> point?
> I think that you are exaggerating how untrustworthy it is, 
> extrapolating from two examples.

These were the FIRST TWO examples.

> I still cannot see your problem with the first of these -- the use of 
> a symbol in the typeset representation and the use of 
> Element[n,Integers] && n > 0 in the code -- and the second formulae, 
> involving ellipsis, is not incorrect, only incomplete.

I do not strongly object to the   "n in positive integers"    versus "n 
in integers  and also n>0".  What I do object to is
the fact that MathML  renders in different NON equivalent ways.  In GIF 
it looks OK.  In Inputform it renders
as  "n ? integers"    when it should be   "n in positive integers"  or 
"n in integers and also n>0".  This is an error, and
I don't care whether you blame MathMLCentral or functions.wolfram.

The second formula is not incomplete, it is wrong.  What does  
Sum[Ellipsis f[i],{i,0,n}] mean?  Is it the
same as Ellipsis Sum[f[i],{i,0,n}], for example?  It is meaningless.

> The point is that this site, even accepting the problems you've 
> identified, is still more useful (at least to someone who does sanity 
> checks on the formulae) than DLMF or ESF. 

I'm glad you find it useful.  I find it disappointing that , even if the 
standardform display is correct, the
CAS version may be different and wrong, and presumably must be checked ALSO.

>> [You may say, just as G&R accept reports of errors, so can the web 
>> site --- yes, this is an important positive aspect of the on-line 
>> repository. Will Michael Trott fix the broken formulas that I have 
>> reported?
> Michael Trott certainly has fixed all the problems that I have reported. 

You are probably reporting the kinds of typos and glitches that fit in 
his design.  I am reporting
the kind of error that requires at least revisiting every formula with 
an ellipsis in it and
a change to the design. 

>> Will he go through all formulas with ellipsis and mark the InputForm 
>> somehow? I hope so.]
> That is up to Michael and the functions team.
> Cheers,
> Paul

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