The discussion revolves around the differentiability of the function f(z) = ¯z in the context of complex analysis. Participants are exploring why this function is not differentiable for any z in the complex plane.
Participants are actively engaging with the problem, considering different approaches to demonstrate the non-differentiability of the function. There is a recognition that the limits yield different results depending on the direction of approach, indicating a productive exploration of the topic.
Some participants note that the Cauchy-Riemann equations were learned later in their studies, which may influence their approach to the problem. There is also an acknowledgment of the definition of the derivative as a viable method for analysis.
Applejacks said:Homework Statement
Show that f(z) = ¯z is not differentiable for any z ∈ C.
Homework Equations
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.
Applejacks said: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?