On 18 November 2014 14:14, Bill Page <bill.p...@newsynthesis.org> wrote:
> 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.
> ...
> 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.
>
Let me add that the kind of solution to this problem that I did
imagine was to implement two derivatives, for example both
f.diff(z) = df/dz + df/d conjugate(z)
and
f.diff2(z) = df/dz - df/d conjugate(z)
diff(z) would equal diff2(z) for all analytic functions and diff would
reduce to the derivative of real non-analytic functions as you desire.
Note that for abs we have
abs(z).diff2(z) = 0
but not in general. There would be no need to discuss this 2nd
derivative with less experienced users until they were ready to
consider more "advanced" mathematics.
Clearly we could implement the chain rule given these two derivatives.
Bill.
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