On 17 November 2014 23:16, Ondřej Čertík <ondrej.cer...@gmail.com> wrote:
> Hi Bill,
>
> Thanks for the clarification. So your point is that 2) is not
> sufficient, that we really need two Wirtinger derivatives --- it's
> just that one can be expressed using the other and a conjugate,
> so perhaps CAS can only return one, but a chain rule needs
> modification and probably some other derivatives handling as
> well. I need to think about this harder.
>
Yes, that is a good summary. My tentative conclusion was that we
could implement just one (Wirtinger) derivative, a modified chain rule
and a sufficiently strong conjugate operation. This derivative is the
same as the usual derivative in the case of analytic functions but we
would have to live with the fact that it is slightly different (factor
of 1/2) for the case of common real derivatives of non-analytic
functions such as abs. Introducing a factor of 2, such as in the case
of the definition of the sign function seems like a small price to
pay.
> Here is a relation that I found today in [1] (see also the references
> there), I don't know if you are aware of it:
>
> D f / D z = df/dz + df/d conjugate(z) * e^{-2*i*theta}
>
> Where Df/Dz is the derivative in a complex plane along the direction
> theta (the angle between the direction and the x-axis) and df/dz and
> df/d conjugate(z) are the two Wirtinger derivatives. This formula
> holds for any function. So all the derivatives no matter which
> direction lie on a circle of radius df/d conjugate(z) and center
> df/dz.
>
> [1] Pyle, H. R., & Barker, B. M. (1946). A Vector Interpretation of
> the Derivative Circle. The American Mathematical Monthly, 53(2), 79.
> doi:10.2307/2305454
http://phdtree.org/pdf/36421281-a-vector-interpretation-of-the-derivative-circle/
Thank you. I was not aware of that specific publication. I think
their geometric interpretation is useful.
>
> For CAS, one could probably just say that theta=0 in our definition,
> and then everything is consistent, and we only have one derivative,
> 2). The other option is to return both derivatives and make the
> derivative Df/Dz of non-analytic function equal to the above formula,
> i.e. depending on df/dz, df/d conjugate(z) and theta.
I think you are overly focused on trying to define a derivative that
reduces to the conventional derivative of non-analytic functions over
the reals.
>
> I need to think about the chain rule. I would simply introduce the
> theta dependence into all formulas, as that gives all possible
> derivatives and gives the exact functional dependence of all
> possibilities. And then see whether we need to keep all formulas
> in terms of theta, or perhaps if we can set theta = 0 for everything.
>
It is not clear to me how to use such as "generic" derivative in the
application of the chain rule.
Bill.
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