Hi Bill, On Thu, Nov 13, 2014 at 10:16 AM, Bill Page <bill.p...@newsynthesis.org> wrote: > It has always seemed very inconvenient to me that "computer algebra > programs such as Mathematica" choose to define derivative as > complex-derivative. I believe a reasonable alternative is what is > known as a Wirtinger derivative. Wirtinger derivatives exist for all > continuous complex-valued functions including non-holonomic functions > and permit the construction of a differential calculus for functions > of complex variables that is analogous to the ordinary differential > calculus for functions of real variables > > http://en.wikipedia.org/wiki/Wirtinger_derivatives > > Wirtinger derivatives come in conjugate pairs but we have > > f(x).diff(conjugate(x)) = conjugate(conjugate(f(x).diff(x)) > > so we really only need one derivative given an appropriate conjugate > function. The Cauchy-Riemann equations reduce to > > f(x).diff(conjugate(x)) = 0 > > I also like that abs is related to the sgn function > > abs(x).diff(x) = x/abs(x) > > This is consistent with > > abs(x)=sqrt(x*conjugate(x)) > > The Wirtinger derivative of abs(x) is 1/2 x/abs(x). Its total > Wirtinger derivative is x/abs(x). > > I have implemented conjugate and Wirtinger derivatives in FriCAS > > http://axiom-wiki.newsynthesis.org/SandBoxWirtinger > > Unfortunately I have not yet been able to convince the FriCAS > developers of the appropriateness of this approach. I would be happy > to find someone with whom to discuss this further, pro and con. The > discussion on the FriCAS email list consisted mostly of the related > proper treatment of conjugate without making explicit assumptions > about variables.
Thanks for your email! I haven't talked to you in a long time. Literally just today I learned about Wirtinger derivatives. The wikipedia page is *really* confusing to me. It took me a while to realize, that Wirtinger derivatives is simply the derivative with respect to z or conjugate(z). I.e. z = x + i*y conjugate(z) = x - i*y >From this it follows: x = 1/2*(z + conjugate(z)) y = i/2*(-z+conjugate(z)) Then I take any function and write it in terms of z and conjugate(z), some examples: |z| = sqrt(z*conjugate(z)) Re z = x = 1/2 * (z + conjugate(z)) z^2 = (x+i*y)^2 And then I simply differentiate with respect to z or conjugate(z). This is called the Wirtinger derivative. So: d|z|/dz = d sqrt(z*conjugate(z)) / dz = 1/2*conjugate(z) / |z| 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? 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.
