z→x3+ i(1 - y)3: Show where the functions is analytic and differentiable.
The Attempt at a Solution
For a function to be analytic cauchy-riemann equations must hold.. so
ux = vy and uy = -vx
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:
ux = 3x2 ; vy = -3(1 - y)2 and
uy = 0 ; -vx = 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.