On Saturday, December 13, 2014 6:08:58 AM UTC-8, Joachim Durchholz wrote: > > Am 13.12.2014 um 06:27 schrieb Richard Fateman: > > > > On Thursday, December 11, 2014 8:16:09 AM UTC-8, Joachim Durchholz > wrote: > >> > >> Am 11.12.2014 um 00:40 schrieb Richard Fateman: > >>> 1994 paper by Adam Dingle and Richard Fateman > >>> Branch Cuts in Computer Algebra, (ISSAC '94 proceedings. also search > >>> online). > >> > >> That paper assumes that everything can be refactored to logarithms plus > >> arithmetic. > >> Does that assumption hold? I could imagine that expressions containing > >> irreducible integrals might not be normalizable in that fashion. > > Repeating the unanswered questions: > > Does the paper assume that everything is normalizable to log/exp? >
I haven't read the paper again, but I don't see why we would assume that everything that one could express in a computer algebra system would be analytic; or entire, if that's the right word. > > If yes, does this assumption hold in general? > As I responded previously, the paper, as well as 99% of the books talk about functions of a (single) complex variable, and are therefore not of much help for functions with additional complex parameters. > > If the assumption does not hold in general, how large is the class of > problems where it does hold, as opposed to all problems that might be > relevant to somebody doing symbolic math? > Well, if you are doing homework problems in an undergraduate or maybe graduate class in complex variables, it might be useful to have a computer program that did some useful things. It also turns out that there are strong connections to "special functions" that could also be made easier if the programs were systematically clean. > > >>> When you say things about sqrt(), does it generalize to cuberoot? If > it > >>> does not, you are in trouble, or will be down the road. > >>> > >>> What is the principal value of (1)^(1/6) ? > > > > Well, there are 6 possible values. > Heh. I didn't write the question, and I knew the answer :-) > > > You could pick (same as sqrt rule?) the positive one, namely 1. > > That doesn't strike me as a good choice, > > I think picking *any* principal root would be problematic, because one > would miss equalities (think x^12 = x^15). > I would pick one of the roots such that its powers cycled through all the other ones, if I had to pick just one... > But that wasn't my point anyway. I was thinking about calculating for > all principal roots in parallel, and not choosing at all until it turns > out that some choice is inconsistent with other assumptions. > This would be, in my opinion, an excellent tactic. > > > since powers of 1 do not generate > > the other roots. > > What's the definition of "generate"? That you can get all principal > roots by taking a power of the chosen value? > yes. But I think you are mistaken in thinking there are 6 principal roots. There are maybe 4? > > Not that I think that property is *that* important, you can always use > polar representation and generate additively (that's more in line with > the approach in your paper anyway). > You can do all kinds of things if you know what you are doing. The CAS is supposed to do the right thing for people who don't know what they are doing. > > >>> You think you can use sqrt because the quadratic formula says > >>> -b+-sqrt(b^2....) > >>> etc. > >>> So you think you know what sqrt means. > >> > >> Not sure what assumptions you assume. > >> > > No assumptions whatsoever. > > You're making assumptions about my knowledge of math if you say "so you > think you know xxx". > I was wondering about these. > True, I am assuming you met up with the quadratic formula. In high school. > > >>> But in that formula you can switch the values +- and the formula > >>> is still valid. > >> > >> Not sure what you mean with that - switching the signs means I still > get > >> the same set of expressions. > > > > That's why the formula works. switching the sign on sqrt just exchanges > the > > roots. > > The point now being...? > As long as you keep the formula together (both roots) it doesn't matter which sign you pick. However, if you separate them and ask which is (say) positive, now it matters. > > >>> How many other formulas do you expect to fiddle with where that is > true? > >>> > >>> Certainly not this one: sqrt(y^2) = abs(y). > >> > >> That could still be handled by doing case distinctions. > > > > If you plot abs(y) you get a V-shaped curve, with a singularity at 0. > > neither square root of y^2 has such a plot. > > Yeah, you need both plots. > And you need to match them with abs() using a case distinction. Which > happen to be the same as the one abs() is making (otherwise the equation > wouldn't hold). > huh? abs(x) is not analytic. It is really a bad idea to introduce such functions if they are not necessary. > > >> This is an interesting case though, since it tells us that we need a > >> quantor: Are we interested in "there exists a branch where the > >> expression holds" or in "the expression holds in all branches"? > >> I think we need the former when determining all solutions to an > >> equation, and the latter when verifying an assumption. > > > > I'm not familiar with the term quantor, > > Oh right, it's "quantifier" in English. > > > but it makes sense to me to > > have 2 separate questions here. > > Agreeing on that one then. > > >>> So if you go off and do the wrong thing, it is probably prudent to > >>> understand that you are doing the wrong thing. > >> > >> Actually the "wrong thing" could be the right thing in specific > >> circumstances (experimental physics is full of this kind of stuff), > > > > perhaps that is why some people advise against learning > > math from a physicist. > > Heh. I can understand that sentiment. > (I think the same sentiment works for learning programming...) RJF -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/d78b9121-bdbf-4334-9e10-486f2f99c852%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
