On 13 November 2014 19:24, Ondřej Čertík <ondrej.cer...@gmail.com> wrote: > On Thu, Nov 13, 2014 at 2:00 PM, Ondřej Čertík <ondrej.cer...@gmail.com> > wrote: >> >> As you said, the function is analytic if it doesn't functionally >> depend on conjugate(z), as can be shown easily. So |z| or >> Re z are not analytic, while z^2 is. If the function is analytic, >> then df/d conjugate(z) = 0, and df/dz is the complex derivative. >> Right? >
Yes. In my email I notice that I wrote "holonomic" but what I meant was "holomorphic". Complex-analytic functions are holomorphic and vice-versa. > ... > So in a CAS, we can simply define the derivative f'(z) as > \partial f / \partial x for any function, even if it doesn't have a > complex derivative. Yes. > For any function we can show that: > > \partial f / \partial x = d f / d z + d f / d conjugate(z) > > Bill, is this what you call the "total Wirtinger derivative"? > Yes > For example, for |z| we get: > > |z|' = \partial |z| / \partial x = d |z| / d z + d |z| / d > conjugate(z) = conjugate(z) / (2*|z|) + z / (2*|z|) = Re(z) / |z| > > Using our definition, this holds for any complex "z". Then, if "z" is > real, we get: > > |z|' = z / |z| > > Which is exactly the usual real derivative. Bill, is this what you had > in mind? That a CAS could return the derivative of abs(z) > as Re(z) / abs(z) ? > > Ondrej > >> >> So for analytic functions, Wirtinger derivative gives the same answer >> as Mathematica. For non-analytic functions, Mathematica leaves it >> unevaluated, but Wirtinger derivative gives you something. >> >> How do you calculate the total Wirtinger derivative? How is that defined? >> >> Because I would like to get >> >> d|x| / d x = x / |x| >> >> for real x. And I don't see currently how is this formula connected to >> Wirtinger derivatives. Finally, the derivative operator in a CAS could >> return Wirtinger derivatives, I think it's a great idea, if somehow we >> can recover the usual formula for abs(x) with real "x". >> >> What are the cons of this approach? >> >> Ondrej > > -- > 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. -- 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.
