Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
analytic_function [2014/02/21 11:46] nikolaj |
analytic_function [2018/03/10 19:52] nikolaj |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ===== Analytic function ===== | ===== Analytic function ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $\mathcal O\subset \mathbb C$ | | + | | @#55CCEE: context | @#55CCEE: $\mathcal O\subset \mathbb C$ | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $f\in \mathrm{it}$ | | |
| - | + | | @#AAFFAA: inclusion | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | | |
| - | | @#FFBB00: $f\in \mathrm{it}$ | | + | | @#FFFDDD: for all | @#FFFDDD: $c$ ... series in $\mathbb C$ | |
| - | + | ||
| - | | @#AAFFAA: $f:\mathcal O\to\mathbb C$ | | + | |
| - | + | ||
| - | | @#FFFDDD: $c$ ... series in $\mathbb C$ | | + | |
| >todo, roughly | >todo, roughly | ||
| 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 | ||
| Line 27: | Line 24: | ||
| {{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 |
| - | $z^n=r^n\,\mathrm{e}^{i\,n\,\theta}=r\,\cos(n\,\theta)+i\,r\,\sin(n\,\theta)$ | + | $z^n=r^n\,\mathrm{e}^{i\,n\,\theta}=r^n\,\cos(n\,\theta)+i\,r^n\,\sin(n\,\theta)$ |
| Both real and imaginary part oscillate along $\theta$. So from the plots below alone it is obvious that the average fence height for $n\neq 0$ must be zero | Both real and imaginary part oscillate along $\theta$. So from the plots below alone it is obvious that the average fence height for $n\neq 0$ must be zero | ||
| Line 55: | Line 52: | ||
| == Cauchy's integral formula == | == Cauchy's integral formula == | ||
| $\frac{1}{n!}f^{(n)}(p) = \frac{1}{2\pi\, i} \oint_\gamma \frac{f(z)}{(z-p)^{n+1}}\, \mathrm dz$ | $\frac{1}{n!}f^{(n)}(p) = \frac{1}{2\pi\, i} \oint_\gamma \frac{f(z)}{(z-p)^{n+1}}\, \mathrm dz$ | ||
| + | |||
| + | Roughly, the Laplace transform uses this for a re-encoding of a functions $f:\mathbb R^+\to\mathbb R$ with Taylor expansion $f(t)=\sum_{n=0}^\infty a_n t^n$, namely by mapping $t^n$ to $s^{-n}\cdot \frac{1}{s}$. | ||
| === 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]] | ||
