Does the critical value exist?

1. Aug 24, 2004

gvk

There is a problem in Arnold's Trivia :yuck::
Find the critical values and critical points of the mapping z -> z^2+ 2*conjugate(z); (sketch the answer).
It seems to me that the critical values (z where f'(z)=0) do not exist, becouse the function f(z)= z^2+ 2*conjugate(z) is not analytic.
Is anybody able to explain what Arnold mean under the notion 'critical values and critical points'?

Thanx for help.

2. Aug 25, 2004

phoenixthoth

if i'm not mistaken, a critical value would also be a place where f' is undefined (in addition to a point where f' is zero).

you can use the cauchy riemann equations but i think you have to see if the function is analytic at the origin using the limit definiton of derivative though i'm not sure.