The function f(z) = ¯z is not differentiable for any z ∈ C due to the failure of the Cauchy-Riemann (CR) equations. The analysis shows that the partial derivatives do not satisfy the necessary conditions: du/dx = 1 ≠ -1 = dv/dy and du/dy = 0 ≠ 0 = -dv/dx. An alternative approach using the definition of the derivative confirms this, as the limit of (f(z+h) - f(z))/h yields different results depending on whether h is real or imaginary. Therefore, f(z) is not analytic anywhere in the complex plane.
PREREQUISITESStudents of complex analysis, mathematicians, and educators seeking to understand the differentiability of complex functions and the implications of the Cauchy-Riemann equations.
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?