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