Perhaps this is more or less where Richardson's theorem enters. http://en.wikipedia.org/wiki/Richardson%27s_theorem
We badly want a reliable way to determine when an expression is identically zero. In general this is not possible, but if we restrict our selves to a subset of "elementary" functions, in particular if we can avoid 'abs', then it is in principle decidable (not withstanding the possible undecidability of equality of constants). As I understand it FriCAS effectively relies on this as part of the machinery for integration, e.g. in 'rischNormalize'. Waldek's challenge to me on the FriCAS list in regards to my proposals related to conjugate and this thread was to show that it is possible to include 'conjugate' and still have a decidable system given the complex equivalent of Richardson's theorem. So far I have not been able to meet this challenge or even to find any specific relevant related publications. Perhaps it is obvious that this is not possible given the definition of abs in terms of conjugate and sqrt. I would be interested in anyone here has considered this issue or might suggest some leads. Of course this is likely not of too much interest in computer algebra systems that take a more pragmatic approach than FriCAS/Axiom. Bill. On 20 November 2014 11:20, Ondřej Čertík <ondrej.cer...@gmail.com> wrote: > > > > > In general, my approach is that I try to define the derivative of > > abs(x) in the simplest possible way, which seems to be in terms of > > abs(x) as well, instead of sqrt(x*conjugate(x)). But the CAS needs to > > be able to rewrite it later if needed, because sometimes things can > > simplify. > > Or to say it with different words, the reason we even have functions > like exp(x), csc(x), sinh(x), asin(x), asinh(x) is that things are > simpler if you use those as opposed to their more elementary function > definition. Perhaps with the exception of csc(x) = 1/sin(x), where I > personally don't see an advantage of introducing a new function for > just 1/sin(x). But with all the other ones, they simplify things, > sometimes. And so the art is to use those in a way to create the most > simple expression at the end. I think that's all there is to it. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
