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analytic_function [2014/02/25 10:25] nikolaj |
analytic_function [2014/11/30 18:38] nikolaj |
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| ===== Analytic function ===== | ===== Analytic function ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $\mathcal O\subset \mathbb C$ | | + | | @#55CCEE: context | @#55CCEE: $\mathcal O\subset \mathbb C$ | |
| - | | @#FFBB00: $f\in \mathrm{it}$ | | + | | @#FFBB00: definiendum | @#FFBB00: $f\in \mathrm{it}$ | |
| - | | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | | + | | @#AAFFAA: inclusion | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | |
| - | | @#FFFDDD: $c$ ... series in $\mathbb C$ | | + | | @#FFFDDD: for all | @#FFFDDD: $c$ ... series in $\mathbb C$ | |
| >todo, roughly | >todo, roughly | ||
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| {{fence_height.png}} | {{fence_height.png}} | ||
| - | A complex function $f(z)$ consists of two real functions, so it's fence height is just given by the sum of the fence heigh of \mathrm{Re}\,f(z)$ and $i\,\mathrm{Im}\,f(z)$. Let's consider the function $f(z):=z=r\,\cos\theta+i\,r\,\sin\theta$. We have | + | A complex function $f(z)$ consists of two real functions, so it's fence height is just given by the sum of the fence heigh of $\mathrm{Re}\,f(z)$ and $i\,\mathrm{Im}\,f(z)$. Let's consider the function $f(z):=z=r\,\cos\theta+i\,r\,\sin\theta$. We have |
| $z^n=r^n\,\mathrm{e}^{i\,n\,\theta}=r\,\cos(n\,\theta)+i\,r\,\sin(n\,\theta)$ | $z^n=r^n\,\mathrm{e}^{i\,n\,\theta}=r\,\cos(n\,\theta)+i\,r\,\sin(n\,\theta)$ | ||
