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Here there are two graphs for two functions from $R^2\mapsto R$.

Is there similar graph for the absolute value of an analytic complex variable function $f:C\mapsto C$ that has the same point (like saddle point or transition). I know some functions that have the point $(x,y,|f(x+iy)|)$ on that such that in one direction it is maximum, and in the other direction it is minimum.

My question here is that: is there any such point such that in one direction it is maximum (or minimum) but in the other direction it is not maximum nor minimum (similar to $(0,0)$ in $y=x^3$ in the real case).


graph 1

graph 2

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thanks to Robert Israel, he answered in mathoverflow. but I like to see an example function and more explanation. –  asd Dec 27 '11 at 21:12
@Robert Israel: Yes, it is. but I was meaning that in one direction similar to $x^2$ and in the other direction similar to $x^3$. For example: $f(z)=\cos(z)$ in $z=0$ has that condition, $|f|$ in one direction is like $x^2$ and in the other is like $(-x^2)$ –  asd Dec 27 '11 at 21:56
If there is some relevant information at mathoverflow you should include a link to that information. –  Gerry Myerson Apr 30 '12 at 4:32

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