On 17 November 2014 15:17, Ondřej Čertík <ondrej.cer...@gmail.com> wrote: > On Sat, Nov 15, 2014 at 9:18 AM, Bill Page <bill.p...@newsynthesis.org> wrote: >> >> I am sorry for the confusion. What I am proposing is that the >> Wirtinger derivative(s) be considered the fundamental case (valid >> for complex or even quaternion variables). As you noted previously >> this is fine and doesn't change anything for the case of analytic >> functions. If someone wants the derivative of a non-analytic >> function over a given domain that should be called something >> else. > > I still don't understand exactly your proposal. We've played with a > few ideas above, in particular we have considered at least (below > d/dz is the Wirtinger derivative, d/dx and d/d(iy) are partial derivatives > with respect to "x" or "iy" in z=x+i*y) : > > 1) d/dz > 2) d/dz + d/d conjugate(z) = d/dx > 3) d/dz - d/d conjugate(z) = d/d(iy) > 4) 2 * (d/dz + d/d conjugate(z)) > 5) 2 * d/dz > > Which of these do you propose to use?
Both d/dz and d/d conjugate(z), i.e. the Wirtinger derivatives. > ... > When "z" is real, then the (real) derivative of |z|' = z/|z|. We want > our complex formula to be equal to z/|z| if "z" is real. Presumably you intend to choose only one of these? But this cannot work in the general case. > ... > As such, only option 2) is consistent. For all analytic functions, it > gives the correct complex derivative, and for non-analytic functions, > at least for abs(z) it reduces to the correct real derivative in the > special case when "z" is real, i.e. z = conjugate(z). Yes. >> ... >> Yes exactly, the concept of "real derivative" is a special case. > > Hopefully the above clarifies, that from everything that we have > considered so far, only the option 2) can work. It turns out that > that's also precisely what also ginac considered for abs(z)'. So > the conclusion seems clear --- simply use 2) for any function, be > it analytic or not. > If there is only one derivative, how will you handle the chain rule? > > However, Bill, from your emails, you seem to be giving conflicting > statements. It seems you agree that 2) is the way to go in some > emails, but then in some other emails you write: > >> It seems to me that we should forget about x and y. All we really >> need is >> >> |z|' = d |z| / d z = conjugate(z) / (2*|z|) > > Which is the case 1) above, and it is shown that it doesn't work. > We need both Wirtinger derivatives. Option 2) is their sum. > Right in the next paragraph you wrote: > >> >> The constant 1/2 is irrelevant. > > What do you mean that the constant 1/2 is irrelevant? I think it is > very relevant, as it makes the answer incorrect. > I said that actually only one Wirtinger derivative was required because the other can be expressed in terms of conjugate but I did not mean to imply that it would meet your criteria of reducing to exactly to the real case. It just happens that both Wirtinger derivatives are the same in the case of abs. > > When you say "I am proposing that the Wirtinger derivative(s) be > considered the fundamental case", which of the five cases above > are you proposing? Strictly speaking, Wirtinger derivative is the case > 1), but that doesn't work. Are you proposing the case 2) instead? > No, I meant that other derivatives (such as the real derivative) can be obtained from the Wirtinger derivatives but not vice versa. > ... >> If someone wants the derivative of a non-analytic function over a >> given domain that should be called something else. > > Are you proposing to only consider analytic functions? No. > I thought the whole conversation in this thread was about how to > extend this to non-analytic functions... Yes. The main issue is non-analytic (non-holomorphic) functions. > ... > I thought the goal was rather to extend the definition of "derivative" > to also apply for non-analytic functions, in the whole complex domain > in such a way, so that it reduces to a complex derivative for analytic > functions, and a real derivative if we restrict "z" to be real. It > seems that 2) above is one such definition that would allow that. > 2) alone is not sufficient. In general for non-analytic functions two derivatives are required. > Bill, would you mind clarifying the above misunderstandings? > I think we are on the same page, probably we both just understood > something else with the terminology we used, but I want to make > 100% sure. > Thank you. I am happy to continue the discussion. Bill. -- 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.
