Taylor series of a complex function that is not holomorphic - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:19:36Z http://mathoverflow.net/feeds/question/11558 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11558/taylor-series-of-a-complex-function-that-is-not-holomorphic Taylor series of a complex function that is not holomorphic Domagoj Peharda 2010-01-12T17:16:34Z 2010-01-12T17:49:15Z <p>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.</p> <p>Bonus question: Can I produce a Taylor series using only derivations over real part?</p> http://mathoverflow.net/questions/11558/taylor-series-of-a-complex-function-that-is-not-holomorphic/11565#11565 Answer by Ben Webster for Taylor series of a complex function that is not holomorphic Ben Webster 2010-01-12T17:35:03Z 2010-01-12T17:35:03Z <p>Remember that the complex plane is $\mathbb{R}^2$ and use <a href="http://en.wikipedia.org/wiki/Taylor%5Fseries#Taylor%5Fseries%5Fin%5Fseveral%5Fvariables" rel="nofollow">normal old multivariable Taylor series</a>.</p> http://mathoverflow.net/questions/11558/taylor-series-of-a-complex-function-that-is-not-holomorphic/11568#11568 Answer by 002 for Taylor series of a complex function that is not holomorphic 002 2010-01-12T17:49:15Z 2010-01-12T17:49:15Z <p>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 $mn=0$.</p>