Question

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? Explain the problem with step by step process?

Answer 1

Consider

$f(z)= |z|^2\\ det \space \Delta w = f(z + \Delta z)- f(z)\\ \frac{ \Delta w}{ \Delta z}=\frac{ |z+\Delta z|^2-|z|^2}{ \Delta z}\\ \frac{ \Delta w}{ \Delta z}=\frac{ (z+\Delta z)(\bar{z}+\bar{\Delta z})-z \bar{z}}{ \Delta z}\\ \frac{ \Delta w}{ \Delta z}=\bar{z}+\bar{\Delta z}+\frac{ z\bar{\Delta z}}{ \Delta z}\\$

When the horizontal line and vertical line approach $\bar{\Delta z}$ towards the origin

$\bar{\Delta z}={\Delta z}, \bar{\Delta z}=-{\Delta z}$

Then

$\frac{\Delta w}{\Delta z}=\bar z+\Delta z+z$ for horizontal line

$\frac{\Delta w}{\Delta z}=\bar z-\Delta z-z$ for vertical line

as $\Delta z \to 0$

$\bar z+\Delta z=\bar z-\bar z$ or z= 0

So f'(0) does not exist when $z \not=0$