# Differences

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 holomorphic_function [2015/01/29 13:05]nikolaj holomorphic_function [2016/03/18 19:20] (current)nikolaj Both sides previous revision Previous revision 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 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 18: Line 18: $f:\mathcal O\to \mathbb C$ ... holomorphic on $\mathcal{O}\ \equiv\ \forall(z_0\in\mathcal O).\ \exists (f':​\mathcal O\to \mathbb C).\ J_{z_0}^f(d)=f'​(z_0)\cdot(d^1+i d^2)$ $f:\mathcal O\to \mathbb C$ ... holomorphic on $\mathcal{O}\ \equiv\ \forall(z_0\in\mathcal O).\ \exists (f':​\mathcal O\to \mathbb C).\ J_{z_0}^f(d)=f'​(z_0)\cdot(d^1+i d^2)$ - This viewpoints ​makes 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 $j_1=\frac{\partial u}{\partial x},​\hspace{.5cm} j_2=-\frac{\partial u}{\partial y}$ $j_1=\frac{\partial u}{\partial x},​\hspace{.5cm} j_2=-\frac{\partial u}{\partial y}$ Line 26: Line 26: ==Cauchy–Riemann equations== ==Cauchy–Riemann equations== - ^ $\frac{ \partial u }{ \partial x } = \frac{ \partial v }{ \partial y }$ ^ + ^ $\dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y }$ ^ - ^ $\frac{ \partial u }{ \partial y } = -\frac{ \partial v }{ \partial x }$ ^ + ^ $\dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x }$ ^ Hence Hence - ^ $f$ ... holomorphic $\implies f'(x+i y)=\frac{\partial ​u}{\partial x}-i \frac{\partial ​u}{\partial y}$ ^ + ^ $f$ ... holomorphic $\implies f'(x+i y)=\left(\dfrac{\partial}{\partial x}-i \dfrac{\partial}{\partial y}\right)u$ ^ So for a holomorphic function $f$, the total change (a complex value) is determined by the real part (or alternatively complex part) of $f$ alone. So for a holomorphic function $f$, the total change (a complex value) is determined by the real part (or alternatively complex part) of $f$ alone. - == non-example == + == Non-example == The [[ε-δ function limit|limit definition]] for a function requires that variation of function values stops being large once you get sufficiently close to the fixed point $0$. The [[ε-δ function limit|limit definition]] for a function requires that variation of function values stops being large once you get sufficiently close to the fixed point $0$.