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--- holomorphic_function [2014/11/03 16:34]
nikolaj
+++ holomorphic_function [2015/01/29 13:05]
nikolaj
@@ Line 34: / Line 34: @@
 So for a holomorphic function $f$, the total change (a complex value) is determined by the real part (or alternatively complex part) of $f$ alone.
+== non-example ==
+The [[ε-δ function limit|limit definition]] for a function requires that variation of function values stops being large once you get sufficiently close to the fixed point $0$.
+For functions like the complex conjugation, $f(z):=\overline{z}$, the finite difference
+$\dfrac{f(z)-f(0)}{z-0}=\dfrac{\overline{z}}{z}={\mathrm e}^{-2i\,\mathrm{arg}(z)}$
+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 $\mathrm{Re}(z)=\frac{1}{2}(z+\overline{z}), \mathrm{Im}(z)=\frac{1}{2i}(z-\overline{z})$, 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 ===
@@ Line 49: / Line 60: @@
 Wikipedia: [[http://en.wikipedia.org/wiki/Holomorphic_function|Holomorphic function]]
 ==== Parents ====
-=== Subset of ===
-[[ℂ valued function]]
 === Context ===
 [[Linear approximation]]
+=== Requirements ===
+[[ε-δ function limit]]
+=== Subset of ===
+[[ℂ valued function]]