[DJR using "&" here] &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Newsgroups: sci.math.num-analysis,sci.math.symbolic From: macsyma@world.std.com (Jeffrey P. Golden) Subject: Re: Maple or Abramowitz who is wrong? Date: Wed, 27 Jan 1999 07:18:47 GMT Reply-To: jpg@macsyma.com > Date: Tue Jan 26 16:23:38 EST 1999 > From: harper@kauri.vuw.ac.nz (John Harper) > Newsgroups: sci.math.num-analysis,sci.math.symbolic > Subject: Re: Maple or Abramowitz who is wrong? > Organization: Victoria University of Wellington, New Zealand > Cache-Post-Path: bats.mcs.vuw.ac.nz!harper@kauri.vuw.ac.nz > X-Cache: nntpcache 2.3.2.1 (see http://www.nntpcache.org/) > > In article <78junf$k4t$1@news.tudelft.nl>, Dany wrote: > >I have asked two things in my first posting, > > > >2) I just use the related formule 7.3.7 page 300 from > > Handbook af mathematical functions , Abramowitz. > >however, as you can see.... > > Clearly A&S are wrong. LHS and middle of 7.3.7 are odd functions of x, > but C2(pi*x^2/2) is an even function, as x^2 = (-x)^2 ! > (I used to tell my students they understood special functions when they > had found an undocumented error in A&S or Gradshteyn&Ryzhik or Prudnikov > et al.) > > John Harper, School of Mathematical and Computing Sciences, > Victoria University, Wellington, New Zealand > e-mail john.harper@vuw.ac.nz phone (+64)(4)471 5341 fax (+64)(4)495 5045 Well, I won't claim to understand special functions, but in the past I reported to sci.math.symbolic on another such odd function / even function bug in A&S: ==================================================================== Article 15165 (7 more) in sci.math.symbolic: Date: 4 Nov 1994 00:59:34 GMT From: jpg@amber.unm.edu (Jeffrey P. Golden) Subject: Is this a typo in A & S handbook? Reply-To: jpg@macsyma.com > Date: 3 Nov 1994 16:29:18 GMT > From: chen@fractal.eng.yale.edu (Richard Q. Chen) > Subject: Is this a typo in A & S handbook? > Organization: Yale University > > In the famous "Handbook of Mathematical Functions" edited > by Abramowitz and Stegun, the formula 7.1.23 on page 298 > shows the asymptotic formula for the error function. > It states that the formula is valid for |arg(z)| < 3Pi/4. > > Is this a typo? I can only prove the validity of the > formula for |arg(z)| < Pi/2. I don't know, but this reminds me of another typo in both A&S and G&R re. erfc. Both A&S 6.5.17 and G&R 8.359.3 are wrong. Here is a tidbit we've added to the development version of Macsyma: ---------------------------------------------------------------------- /* For x real, convert erf(x) to an expression involving the Incomplete Gamma Function gamma(a,x) = integral of exp(-t)*t^(a-1) wrt t from x to infinity : */ (c1) makegamma(erf(x)); 1 2 gamma(-, x ) 2 (d1) (1 - ------------) signum(x) sqrt(%pi) ---------------------------------------------------------------------- By omitting the signum(x), A&S and G&R are equating an odd function (erf(x)) with an even function (expression in terms of Gamma and x^2 .) ======================================================================== I've reported on at least one other serious bug in G&R as well. I'll repeat that here because amusingly that one also involves John Harper and Keith Geddes! : ======================================================================== Article 25163 (14 more) in sci.math.symbolic: Date: 10 Jan 1997 21:22:05 GMT From: jpg@math.math.unm.edu (Jeffrey P. Golden) Subject: tables of integrals and errors Reply-To: jpg@macsyma.com > Date: 9 Jan 1997 23:42:25 GMT > From: harper@kauri.vuw.ac.nz (John Harper) > Subject: tables of integrals and errors > Organization: Victoria University of Wellington, New Zealand > NNTP-Posting-Host: kauri.vuw.ac.nz > > [...] > > Erdelyi once said "All books of tables like these contain errors. > Don't believe anything you see in them: use it as an indication that > some formula vaguely resembling the one in the book is true, which > it is now your job to find." I tell my students they really understand > special functions when they have found an undocumented error in G+R > (or PBM or Erdelyi or Abramowitz & Stegun). When we implemented a Fourier transform package for Macsyma we compared the Fourier transform results Macsyma got with those in G&R 17.23 Table of Fourier transform pairs. The one for sin(a*x^2) did not agree: Macsyma has a cos where the table in G&R has a sin: f(x) = sin(a*x^2) F(w) = 1/sqrt(2*a)*trigfun(w^2/(4*a)+pi/4) where trigfun = cos for us ; = sin for G&R . So I tried definite integration directly in Macsyma and the results were the same. When I mentioned this to Prof. Keith Geddes of U. of Waterloo he referred me to G&R 3.691.5 which is correct, so G&R doesn't agree with G&R! I sent email on this to the G&R folks but never got a reply. I hope they fix this some day! > [...] > > John Harper School of Math+Comp Sci Victoria Univ Wellington New Zealand =========================================================================== From: Jeffrey P. Golden Organization: Macsyma Inc. Reply-To: jpg@macsyma.com URL: http://www.macsyma.com