Answer in Chemical Engineering for pavani #231353
March 11th, 2023
13)Show that the function f (z)=(z̅ )²/z,z≠0 ;0, z=0 satiesfies Cauchy-Rieman equations at z=0.Does f'(0) exist?
We can prove this using the theorem:
We let f(z) = u + iv be an analytic function.
1. If f 0 (z) is identically zero, then f(z) is a constant.
2. If either Re f(z) = u or Im f(z) = v is constant, then f(z) is constant. In particular, a non-constant analytic function cannot take only real or only pure imaginary values.
3. If |f(z)| is constant or arg f(z) is constant, then f(z) is constant.
,If, f’(z) = 0, then:
Therefore, ∂u/ ∂x = ∂v/ ∂x = 0. By the Cauchy-Riemann equations, ∂v ,∂y = ∂u /∂y = 0
We determine that f (z) is a constant. It proves f’ (0) exists.