show that the following functions are differentiable everywhere and then also find f'(z) and f''(z).
(a) f(z) = iz + 2
so f(z) = ix -y +2
then u(x,y) = 2-y, v(x,y) = x
Cauchy-Riemann conditions says is differentiable everywhere if :
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
The Attempt at a Solution
so using the Cauchy-Riemann conditions i find that the function is differentiable everywhere. the part im stuck on is finding the first derivative.
f'(z) should be in the form of two partial derivatives right? because of the way the variables are set up.
∂z/∂x = ∂z/∂u(∂u/∂x) + ∂z/∂v(∂v/∂x)
∂z/∂y = ∂z/∂u/(∂u/∂y) + ∂z/∂v(∂v/∂y)
but where do i go from here? i can solve partials of u with respect to x or y but i dont know how to solve the partials of z with respect to u.