## Taylor series of a complex function that is not holomorphic

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I want to create Taylor series of a complex function that has complex conjugate in it. Obviously I cannot do a total derivative but derivations over real and imag parts exist.

Bonus question: Can I produce a Taylor series using only derivations over real part?

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Maybe there is something about Taylor series using directional derivatives instead of partial derivatives.  – Domagoj Peharda Jan 13 2010 at 10:39
The reason why I wanted to find a Taylor series was to produce a Newton method. Finally I found that I needed to look into CR Calculus citeseerx.ist.psu.edu/viewdoc/… – Domagoj Peharda Feb 16 2010 at 21:26

## 2 Answers

Another option is $\sum c_{mn}z^m \bar z^n$, which still keeps track of the complex structure. For instance, harmonic functions will have $c_{mn}=0$ unless $mn=0$.

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So I can always transform my complex function to f(z,z*). But how do I do a derivative of that? – Domagoj Peharda Jan 13 2010 at 10:51

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Remember that the complex plane is $\mathbb{R}^2$ and use normal old multivariable Taylor series.

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My function is thus R^2 to R^2 and it seems that your link is only about scalar valued multivariable function.  – Domagoj Peharda Jan 13 2010 at 10:29
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