graph of the size of a complex function - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-19T03:09:08Z http://mathoverflow.net/feeds/question/84405 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84405/graph-of-the-size-of-a-complex-function graph of the size of a complex function asd 2011-12-27T19:17:02Z 2011-12-28T13:00:52Z <p>Hi Here there are two graphs for two functions from \$R^2\mapsto R\$.</p> <p>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. </p> <p>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).</p> <p>Thanks</p> <p><a href="http://postimage.org/image/v2ig8ycx7" rel="nofollow">link text</a></p> <p><a href="http://postimage.org/image/57zgroy4t" rel="nofollow">link text</a></p> http://mathoverflow.net/questions/84405/graph-of-the-size-of-a-complex-function/84410#84410 Answer by Robert Israel for graph of the size of a complex function Robert Israel 2011-12-27T20:33:52Z 2011-12-27T20:33:52Z <p>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\$.</p>