# Taylor series of a complex function that is not holomorphic

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 '10 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 '10 at 21:26

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 $mn=0$.
Remember that the complex plane is $\mathbb{R}^2$ and use normal old multivariable Taylor series.