Question

Verify whether f(z) = z² is analytic or not.

Answer 1

Answer:

If z=x+iy

we have that

f(z)=|z|2=z⋅z¯¯¯=x2+y2

This shows that is a real valued function and can not be analytic.

We can rewrite the above as

f(z)=x2+y2+i⋅0

Set

u(x,y)=x2+y2

v(x,y)=0

Hence

f(x,y)=u(x,y)+i⋅v(x,y)

The function f is continuous because u,v are continuous. But Cauchy Riemann holds at the origin

ux=2x,uy=2y

ux=vy,uy=−vx

x=0,y=0=>z=0

Hence f is differentiable only at the origin, and the derivative is zero.