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.
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.
For analytic functions, we have df/d conjugate(z) = 0, and so the
above formula proves that all the derivatives are independent of
direction theta and equal to df/dz.
For non-analytic functions, the above formula gives all the possible
derivatives, and besides df/dz, the derivatives also depend on df/d
conjugate(z) and theta. But that's it. So as you said, the two
Wirtinger derivatives allow us to calculate the derivative along any
direction theta we want.
>From my last email, the case 1) corresponds to df/d conjugate(z)=0,
i.e. analytic functions and the result is independent of theta. Case
2) is theta = 0, pi, 2*pi, ..., i.e. taking the derivative along the
x-axis. Case 3) is theta = pi/2, 3*pi/2, 5*pi/2, ..., i.e. taking the
derivative along the y-axis.
A real derivative of a real function g(x) is simply taken along the
x-axis. You can imagine that g(x) is also (arbitrarily) defined in the
whole complex plane and you are taking the Dg/gz derivative above with
theta = 0. The result is the same. So that's why the case 2), i.e.
theta=0, always reproduces the real derivative, because real
derivative is defined as theta=0.
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 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.
Ondrej
[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
--
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.
