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 I am 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 don't know how to solve the partials of z with respect to u.