2008/6/1 Henryk Trappmann <[EMAIL PROTECTED]>: > >> there is an "obvious" convention that by default we mean the positive >> root. > > We have to distinguish between solutions of polynomials and roots. > Roots are clearly defined mono-valued functions: > z.nth_root(n)=e^(log(z)/n) > however this function is not continuous in z, as log is not continuous > at the negative real axis. This makes things complicated. > >> It's a lot more complicated when you deal with general >> algebraic numbers which have several ways of being embedded into CC. >> Even for square roots of negative reals: you might suggest taking the >> root with positive imaginary part, but then sqrt(-2)*sqrt(-3) equals >> -sqrt(6) and not +sqrt(6). > > by the above definition this can easily be computed: > sqrt(-2)*sqrt(-3)=e^(log(-2)/2+log(-3)/2)=sqrt(6)e^(-pi*i/2-pi*i/ > 2)=sqrt(6)*(-1) >
You have only shifted the ambiguity to the multi-values proprty of log! As I have often expleined to students, the "generic" proprty that log(ab)=log(a)+log(b) just does not hold as an identity for arbitrary complex numbers, and there is no branch convention which will make that true. > I never said it is simple but I am sure that there are equality > deciding algorithms. > And I really want to learn about those. If they exist, I would like to see them too. But I am not optimistic. However there must be a vast literature on the subject, which is at least as old as computer algebra. John > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
