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holomorphic_function [2015/01/29 13:05] nikolaj |
holomorphic_function [2016/03/18 19:20] 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 $0$. | ||
| For functions like the complex conjugation, $f(z):=\overline{z}$, the finite difference | For functions like the complex conjugation, $f(z):=\overline{z}$, the finite difference | ||
| - | $\dfrac{f(z)-f(0)}{z-0}=\dfrac{\overline{z}}{z}={\mathrm e}^{-2i\mathrm{arg}(z)}$ | + | $\dfrac{f(z)-f(0)}{z-0}=\dfrac{\overline{z}}{z}={\mathrm e}^{-2i\,\mathrm{arg}(z)}$ |
| varies strongly with the $z$'s argument, even if $z$ is varied in only an arbitrarily small area around $0$, and so there is no limit. The function $f$ doesn't have a derivative. | varies strongly with the $z$'s argument, even if $z$ is varied in only an arbitrarily small area around $0$, and so there is no limit. The function $f$ doesn't have a derivative. | ||
