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analytic_function [2016/09/28 18:20]
nikolaj
analytic_function [2018/03/10 19:52] (current)
nikolaj
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 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
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