Z (conjugate) not analytic?

by Applejacks
Tags: analytic, conjugate
 P: 33 1. The problem statement, all variables and given/known data Show that f(z) = ¯z is not differentiable for any z ∈ C. 2. Relevant equations 3. The attempt at a solution Is it because the Cauchy-Reimann Equations don't hold? Z (conjugate) = x-iy u(x,y)=x v(x,y=-iy du/dx=1≠dv/dy=-1 du/dy=0≠-dv/dx=0 Edit: Is there another approach? Because the CR Equations is something we learned later on.
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P: 25,228
 Quote by Applejacks 1. The problem statement, all variables and given/known data Show that f(z) = ¯z is not differentiable for any z ∈ C. 2. Relevant equations 3. The attempt at a solution Is it because the Cauchy-Reimann Equations don't hold? Z (conjugate) = x-iy u(x,y)=x v(x,y=-iy du/dx=1≠dv/dy=-1 du/dy=0≠-dv/dx=0 Edit: Is there another approach? Because the CR Equations is something we learned later on.
Sure. Use the definition of f'(z)=lim h->0 (f(z+h)-f(z))/h. Show the limit is different if you pick h to be real from the limit if you pick h to be imaginary. That's really what the content of the CR equations is.
 P: 33 I think I get it now. (f(z+h)-f(z))/h (conjugate((z+h)-z))/h = h(conjugate)/h If h=Δx, the ratio equals 1 If h=Δiy, the ratio equals -1. Since the two approaches don't agree for any z, z(conj) is not analytic anywhere. Correct?