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holomorphic_function [2015/01/29 13:04] nikolaj |
holomorphic_function [2015/01/29 13:05] nikolaj |
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| varies strongly with the $z$'s argument, even if $z$ is varied in only an arbitrarily small area around $0$, and so there is no limit. The function $f$ doesn't have a derivative. | varies strongly with the $z$'s argument, even if $z$ is varied in only an arbitrarily small area around $0$, and so there is no limit. The function $f$ doesn't have a derivative. | ||
| - | Since $\mathrm{Re}(z)=\frac{1}{2}(z+\overline{z}), \mathrm{Im}(z)=\frac{1}{2}(z-\overline{z})$, this translated to the projections to the real and imaginary axis, and in turn to all functions which can only be defined in terms of those (as oppsed to be defined in terms of $z$ itself, e.g. $z^3$). | + | Since $\mathrm{Re}(z)=\frac{1}{2}(z+\overline{z}), \mathrm{Im}(z)=\frac{1}{2i}(z-\overline{z})$, this translated to the projections to the real and imaginary axis, and in turn to all functions which can only be defined in terms of those (as oppsed to be defined in terms of $z$ itself, e.g. $z^3$). |
| === Theorems === | === Theorems === | ||
