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 analytic_function [2016/09/28 18:20]nikolaj analytic_function [2018/03/10 19:52]nikolaj Both sides previous revision Previous revision 2018/03/10 19:52 nikolaj 2016/09/28 18:20 nikolaj 2014/11/30 18:38 nikolaj 2014/03/21 11:11 external edit2014/02/25 10:25 nikolaj 2014/02/21 11:46 nikolaj 2014/02/21 11:45 nikolaj 2014/02/21 11:31 nikolaj 2014/02/21 11:30 nikolaj 2014/02/21 11:29 nikolaj 2014/02/21 11:29 nikolaj 2014/02/21 11:25 nikolaj 2014/02/21 11:22 nikolaj 2014/02/21 11:21 nikolaj 2014/02/21 11:21 nikolaj 2014/02/21 11:19 nikolaj 2014/02/21 11:18 nikolaj 2014/02/21 11:16 nikolaj 2014/02/21 11:16 nikolaj 2014/02/21 11:15 nikolaj 2014/02/21 11:11 nikolaj 2014/02/21 11:09 nikolaj 2014/02/21 11:07 nikolaj 2014/02/21 11:06 nikolaj 2014/02/21 11:04 nikolaj 2014/02/20 19:58 nikolaj 2014/02/20 19:29 nikolaj 2018/03/10 19:52 nikolaj 2016/09/28 18:20 nikolaj 2014/11/30 18:38 nikolaj 2014/03/21 11:11 external edit2014/02/25 10:25 nikolaj 2014/02/21 11:46 nikolaj 2014/02/21 11:45 nikolaj 2014/02/21 11:31 nikolaj 2014/02/21 11:30 nikolaj 2014/02/21 11:29 nikolaj 2014/02/21 11:29 nikolaj 2014/02/21 11:25 nikolaj 2014/02/21 11:22 nikolaj 2014/02/21 11:21 nikolaj 2014/02/21 11:21 nikolaj 2014/02/21 11:19 nikolaj 2014/02/21 11:18 nikolaj 2014/02/21 11:16 nikolaj 2014/02/21 11:16 nikolaj 2014/02/21 11:15 nikolaj 2014/02/21 11:11 nikolaj 2014/02/21 11:09 nikolaj 2014/02/21 11:07 nikolaj 2014/02/21 11:06 nikolaj 2014/02/21 11:04 nikolaj 2014/02/20 19:58 nikolaj 2014/02/20 19:29 nikolaj 2014/02/20 19:28 nikolaj 2014/02/20 19:12 nikolaj 2014/02/20 19:12 nikolaj old revision restored (2014/02/20 19:08) Line 26: Line 26: 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