# Differences

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 holomorphic_function [2019/10/06 23:33]nikolaj [Discussion] holomorphic_function [2019/10/06 23:37] (current)nikolaj [Discussion] Both sides previous revision Previous revision 2019/10/06 23:37 nikolaj [Discussion] 2019/10/06 23:33 nikolaj [Discussion] 2016/03/18 19:20 nikolaj 2015/01/29 13:05 nikolaj 2015/01/29 13:05 nikolaj 2015/01/29 13:04 nikolaj 2014/11/03 16:34 nikolaj 2014/03/21 11:11 external edit2014/02/20 19:08 nikolaj 2014/02/20 19:01 nikolaj 2014/02/20 18:51 nikolaj 2014/02/20 18:46 nikolaj 2014/02/20 18:43 nikolaj 2014/02/20 18:34 nikolaj old revision restored (2014/02/13 16:13)2013/05/14 23:25 nikolaj removed2013/05/05 22:50 nikolaj 2013/05/05 15:15 nikolaj created 2019/10/06 23:37 nikolaj [Discussion] 2019/10/06 23:33 nikolaj [Discussion] 2016/03/18 19:20 nikolaj 2015/01/29 13:05 nikolaj 2015/01/29 13:05 nikolaj 2015/01/29 13:04 nikolaj 2014/11/03 16:34 nikolaj 2014/03/21 11:11 external edit2014/02/20 19:08 nikolaj 2014/02/20 19:01 nikolaj 2014/02/20 18:51 nikolaj 2014/02/20 18:46 nikolaj 2014/02/20 18:43 nikolaj 2014/02/20 18:34 nikolaj old revision restored (2014/02/13 16:13)2013/05/14 23:25 nikolaj removed2013/05/05 22:50 nikolaj 2013/05/05 15:15 nikolaj created Line 16: Line 16: Let's compare this with the $2\times 2$ Jacobi matrix resulting when computing the derivative of a function taking values in $\mathbb R^2$, to see if the linear approximation at a point $z_0=x_0+i y_0$ of a complex function along $d=\langle d^1,​d^2\rangle$ can be formulated as a multiplication within $\mathbb C$, i.e. by multiplying a $d$-independent function $f'$ with the number $d^1+i d^2$. Given the similarity with the real function derivative, it's not too surprising that this condition is equivalent to the existence of the limit in the definition above. So we could also say Let's compare this with the $2\times 2$ Jacobi matrix resulting when computing the derivative of a function taking values in $\mathbb R^2$, to see if the linear approximation at a point $z_0=x_0+i y_0$ of a complex function along $d=\langle d^1,​d^2\rangle$ can be formulated as a multiplication within $\mathbb C$, i.e. by multiplying a $d$-independent function $f'$ with the number $d^1+i d^2$. Given the similarity with the real function derivative, it's not too surprising that this condition is equivalent to the existence of the limit in the definition above. So we could also say - $f:\mathcal O\to \mathbb C$ ... holomorphic on $\mathcal{O}\ \equiv\ ​ \exists (f':\mathcal O\to \mathbb C).\ \forall(z_0\in\mathcal O).\ J_{z_0}^f(d)=f'(z_0)\cdot(d^1+i d^2)$ + $f:\mathcal O\to \mathbb C$ ... holomorphic on $\mathcal{O}\ \equiv\ ​ \exists (g:\mathcal O\to \mathbb C).\ \forall(z_0\in\mathcal O).\ J_{z_0}^f(d)\overset{\sim}{=}g(z_0)\cdot(d^1+i d^2)$ + + where then $g=f'​$. This viewpoints leads us directly to a very important property of holomorphic functions. Comparing the first components tells us that This viewpoints leads us directly to a very important property of holomorphic functions. Comparing the first components tells us that