A sneaky workaround that I have found works rather well when exponentials to a negative base appear in something you want to be a real valued function. First factor out -1^n from your -b^n expression: -b^n = -1^n * b^n and then let -1^n -= cos(Pi*n) . A bit of basic calculus will show that cos(Pi*n) = the real part of -1^n and is always single valued and analytic,.
^,..,^ On Friday, June 20, 2014 8:21 PM, Gregory Bard <[email protected]> wrote: (Note: I have a lot of cube-root related emails to reply to today. I promise that I will get to everyone, eventually.) For Gerald Smith's point, I think we have to clarify between nth_real_root( x, n ) having domain R x Z+ and range R (which I think is okay) and the undesirable option of nth_real_root( x, n ) having domain R x Q and range C (which I think is inadvisable). If n is a positive integer, then there should be no ambiguity. (See my response to Michel Paul if that's unclear.) It is coherently conceivable to consider nth_real_root( x, n ) having domain R x Q* where we could evaluate nth_real_root( x, p/q ) as ( nth_real_root( x, q ) )^p or equivalently ( nth_real_root( x^p, q ) ). So long as q=/=0, we can use the odd/even status of q to figure out how to handle the situation. However, as Gerald Smith so excellently points out, in each neighborhood of any negative rational value of n, no matter how small, there are some rational numbers with even denominators and some rational numbers with odd denominators. Thus the output would flip-flop between entirely real and entirely imaginary numbers. I guess that's not mathematically forbidden but the word "pathological" is perhaps not uncalled for. Thoughts? ---Greg On Fri, Jun 20, 2014 at 12:47 PM, 'Gerald Smith' via sage-edu <[email protected]> wrote: > Guys, please keep in mind that exponentials of negative bases are not really > functions. They give a mix of real and multivalued complex results. You > cannot treat x^(1/3) as a synonym for cuberoot (x) if x,0. The latter is a > well defined function and the former is not. Perhaps this might be a good > reason to introduce a new function cuberoot(x) and evaluate it (using for > example Newton''s Formula). > Cheers, > MathBear > P.S. If you let the independent variable be the exponent and have a negative > base, then for any interval of x however short, the function will flip flop > infinitely often between real and multivalued complex results. The output is > completely discontinuous and very messy. Just try graphing it! There is a > good reason why exponential finctions of a real variable are restricted to > positive bases. > > > On Thursday, June 19, 2014 11:39 AM, David Smith <[email protected]> wrote: > > > I could live with nthroot -- but is cuberoot (or curt or cbrt) so different > from sqrt? Anyway, it would be nice to have functions that resemble ones > students see routinely without generating error messages. I'm very much > aware of the difficulties involved in reaching that goal, and I'll be > patient. > > Still waiting for a solution on arcsec, as well. There the issue is > primarily plotting, although "integral" fails, even with a proper domain. > numerical_integral works as it should. > > In the great scheme of things, these are certainly peripheral issues -- and > they wouldn't be issues at all if textbook authors didn't feel obliged to > drag in every function they know about. As author, I'll guilty of that too > (fear of the adoption committee), but in my defense, we only do these things > in exercises. > > On Wednesday, June 18, 2014 8:48:36 PM UTC-4, kcrisman wrote: > > Just to chime in, as someone who has dealt with this question a lot (though, > perhaps ironically, never in a classroom situation): > > I would be very against a "cuberoot" function, but an "nthroot" function > where it was really clear what input was allowed could fly. I appreciate > Greg's rationale. Note however - what is the 0.1 power? Is that the same as > the 1/10 power? This is a tricky floating point question to interpret. > > I don't think that the slowdown would be too bad since it would primarily be > for pedagogical purposes for plotting. > > David, I don't know how this would work with integrals, though - we'd have > to see if Maxima had something equivalent. Perhaps it could do a temporary > set of the Maxima domain to real somehow, if that is what allows Maxima to > do the "right thing" in this context, I don't know. > > Thanks for fighting the good fight on trying to resolve this once and for > all! > - kcrisman > > -- > You received this message because you are subscribed to the Google Groups > "sage-edu" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-edu. > For more options, visit https://groups.google.com/d/optout. > > > -- > You received this message because you are subscribed to a topic in the > Google Groups "sage-edu" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/sage-edu/0vGw_Pd6v_Y/unsubscribe. > To unsubscribe from this group and all its topics, send an email to > [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-edu. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-edu" group. 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