On 18 November 2014 13:41, Ondřej Čertík <ondrej.cer...@gmail.com> wrote:
> On Tue, Nov 18, 2014 at 11:08 AM, Bill Page <bill.p...@newsynthesis.org>
> wrote:
>> ...
>> Have you had a chance to consider the issue of the chain-rule yet?
>
> Yes. Very straightforward, as I suggested in my last email. Just start with:
>
> D f / D z = df/dz + df/d conjugate(z) * e^{-2*i*theta}
>
> and then consider the chain rule for Wirtinger derivatives
> (http://en.wikipedia.org/wiki/Wirtinger_derivatives#Functions_of_one_complex_variable_2),
> I am sure that can be proven quite easily.
Let me make sure I understand your proposal. Are you saying that you
would introduce the symbolic expression
e^{-2*i*theta}
with theta undefined in the result of all derivatives? So that
diff(x) is always the sum of two terms. In particular
abs(x).diff(x)
would return the symbolic expression
conjugate(x)/(2*abs(x)) + conjugate(x)/(2*abs(x))* e^{-2*i*theta}
If you are, then clearly one can recover both Wirtinger derivatives
from this expression and the rest holds.
> Then you just calculate directly:
> ...
> So it exactly agrees, except that there is a theta dependence in the
> final answer and GiNaC implicitly chose theta=0.
>...
> I hope I didn't make some mistake somewhere, but it looks all
> straightforward to me.
>
It looks OK to me but I must say, it probably seems rather peculiar
from the point of view expressed earlier by David Roe.
How can you explain the presence of the e^theta term to someone
without experience in complex analysis or at least multi-variable
calculus?
I thought rather that what you were proposing was to set theta=0 from
the start. If you did that, then I think you still have problems with
the chain rule.
Bill.
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