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

Is there similar graph for the absolute value of a 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).


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closed as too localized by Igor Rivin, Alain Valette, George Lowther, Ryan Budney, Yemon Choi Dec 27 '11 at 20:33

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Without further restrictions on your function $f$ (analytic? anti-analytic?) the answer is that a function $C \to R$ is exactly the same as a function $R^2\to R$. – Yemon Choi Dec 27 '11 at 20:35
$f$ is analytic – asd Dec 28 '11 at 14:40
up vote 1 down vote accepted

For any nonconstant analytic function $f$, if $f'(p) = 0$ but $f(p)$ and $f''(p)$ are nonzero, then the graph of $|f(z)|$ will have a saddle point at $p$.

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@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 28 '11 at 14:42
Is it possible $f'(z)\neq0$ and $f(z)\neq0$, but $z$ is saddle point of the above type. – asd Dec 28 '11 at 14:46
If $f'(p) \ne 0$, it's not a saddle point. The definition of a saddle point requires a stationary point. – Robert Israel Dec 28 '11 at 19:26
@Robert Israel: thanks – asd Dec 29 '11 at 10:57

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