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holomorphic_function [2015/01/29 13:05] nikolaj |
holomorphic_function [2019/10/06 23:33] nikolaj [Discussion] |
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| 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\ \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\ \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) $ |
| - | 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 $0$. | ||
