# Complex analysis. Compute bar derivative

Can sage do complex analysis? I was unable to find documentation upon a quick web search.

For instance, I would like to define $f(z) = z \bar{z}$ and ask sage to compute $\displaystyle \frac{\partial f}{\partial \bar{z}}$, is that possible?

Of course I could treat $z$ and $\bar{z}$ as independent variables, but that is not what I'm asking.

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http://en.wikipedia.org/wiki/Wirtinge...

I would agree that this is not implemented in Sage but I would disagree that it can be defined as a "simple combination of the usual derivatives". In particular it is necessary to consider the chain-rule. I have a prototype implementation in FriCAS (Axiom) that does this. In that context I am interested in collaboration and critical review.

http://axiom-wiki.newsynthesis.org/Sa...

From the point of view of complex analysis you might also want to look up references to polygenic functions, e.g. Kasner

http://www.ncbi.nlm.nih.gov/pmc/artic...

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See this discussion for some of the problems that currently arise when trying to do this with non-analytic functions.

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I'm not going to read the discussion in detail, sorry. I'm not sure it's relevant. I don't see how there could be a problem with defining the z-derivative and zbar-derivative, who are simple combinations of the usual partial derivatives. So I'm simply understanding that these are not implemented in sage, oh well.

I think the point is that one would first have to know that z is not a single variable, but really to be treated as a two-variable setup in the first place, as opposed to a single (possibly complex) variable. In which case one is basically doing independent variables anyway.