Show where the functions is anlaytic and differentiable

[Rubik]

Homework Statement

z→x³+ i(1 - y)³: Show where the functions is analytic and differentiable.

Homework Equations

The Attempt at a Solution

For a function to be analytic cauchy-riemann equations must hold.. so
u_x = v_y and u_y = -v_x

Now f(z) = x³ + i(1 - y)³ is already in the form u(x,y) + iv(x,y) with u(x,y) = x³ and v(x,y) = (1 - y)³ So:

u_x = 3x² ; v_y = -3(1 - y)² and
u_y = 0 ; -v_x = 0

Does this then mean cauchy-riemann does not hold anywhere but 0 so f is not differentiable on a neighbourhood of any point in C which means it is nowhere analytic.

 

[Office_Shredder]

From what you wrote down it doesn't look like the Cauchy Riemann equations hold at x=y=0