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analytic_function [2014/03/21 11:11] 127.0.0.1 external edit |
analytic_function [2016/09/28 18:20] nikolaj |
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| ==== Set ==== | ==== Set ==== | ||
| | @#55CCEE: context | @#55CCEE: $\mathcal O\subset \mathbb C$ | | | @#55CCEE: context | @#55CCEE: $\mathcal O\subset \mathbb C$ | | ||
| - | |||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $f\in \mathrm{it}$ | | | @#FFBB00: definiendum | @#FFBB00: $f\in \mathrm{it}$ | | ||
| - | |||
| | @#AAFFAA: inclusion | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | | | @#AAFFAA: inclusion | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | | ||
| - | |||
| | @#FFFDDD: for all | @#FFFDDD: $c$ ... series in $\mathbb C$ | | | @#FFFDDD: for all | @#FFFDDD: $c$ ... series in $\mathbb C$ | | ||
| Line 14: | Line 10: | ||
| >$\exists c.\ \forall z.\ f(z)=\sum_{n=-\infty}^\infty c_n\,z^n$ | >$\exists c.\ \forall z.\ f(z)=\sum_{n=-\infty}^\infty c_n\,z^n$ | ||
| - | ==== Discussion ==== | + | ----- |
| + | === Discussion === | ||
| Picture a continuous function $f:\mathbb R^2\to\mathbb R$ as a surface given by $f(x,y)$ and imagine drawing a circle of radius $1$ around the point at the origin and is parametrized by $\langle \cos\theta,\sin\theta,h\rangle$ where $\theta\in[0,2\pi)$ and $h\in[0,f(\cos\theta,\sin\theta))$. See the picture below. It looks like a cylinder cut of at height $f(\cos\theta,\sin\theta)$. Let's call it a "fence". What is the surface of that fence? Clearly, it's given by the integral $\int_0^{2\pi}\mathrm{d}\theta$ of $f(\cos\theta,\sin\theta)$. And so the average height (if we count negative height as negative contributions) of the fence is | Picture a continuous function $f:\mathbb R^2\to\mathbb R$ as a surface given by $f(x,y)$ and imagine drawing a circle of radius $1$ around the point at the origin and is parametrized by $\langle \cos\theta,\sin\theta,h\rangle$ where $\theta\in[0,2\pi)$ and $h\in[0,f(\cos\theta,\sin\theta))$. See the picture below. It looks like a cylinder cut of at height $f(\cos\theta,\sin\theta)$. Let's call it a "fence". What is the surface of that fence? Clearly, it's given by the integral $\int_0^{2\pi}\mathrm{d}\theta$ of $f(\cos\theta,\sin\theta)$. And so the average height (if we count negative height as negative contributions) of the fence is | ||
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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 |
| $z^n=r^n\,\mathrm{e}^{i\,n\,\theta}=r\,\cos(n\,\theta)+i\,r\,\sin(n\,\theta)$ | $z^n=r^n\,\mathrm{e}^{i\,n\,\theta}=r\,\cos(n\,\theta)+i\,r\,\sin(n\,\theta)$ | ||
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| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Holomorphic_functions_are_analytic|Analyticity of holomorphic functions]] | Wikipedia: [[http://en.wikipedia.org/wiki/Holomorphic_functions_are_analytic|Analyticity of holomorphic functions]] | ||
| - | ==== Parents ==== | + | |
| + | ----- | ||
| === Equivalent to === | === Equivalent to === | ||
| [[Holomorphic function]] | [[Holomorphic function]] | ||
| === Requirements === | === Requirements === | ||
| [[Infinite sum of complex numbers]] | [[Infinite sum of complex numbers]] | ||
