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holomorphic_function [2015/01/29 13:05]
nikolaj
holomorphic_function [2016/03/18 19:20] (current)
nikolaj
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 $f:\mathcal O\to \mathbb C$ ... 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 
  
 $j_1=\frac{\partial u}{\partial x},​\hspace{.5cm} j_2=-\frac{\partial u}{\partial y}$  $j_1=\frac{\partial u}{\partial x},​\hspace{.5cm} j_2=-\frac{\partial u}{\partial y}$ 
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 ==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 $0$.
  
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