yes, you made a mistake with dv/dy though, it is x(ii) f(z) = xy + xyi
Then u(x,y) = xy and v(x,y) = xy.
du/dx = y
du/dy = x
dv/dx = y
dv/dy = x
This means x = y and x = -y and f(z) is only differentiable along these lines. So it's differentiable at points z(x,x) and z(x,-x) and its derivative is df/dx = y + yi = x + xi and x - xi at these points.
nice(iii) f(z) = y- 2xy + i(-x + x^2 - y^2).
Then u(x,y) = y- 2xy and v(x,y) = -x + x^2 - y^2
du/dx = -2y
du/dy = 1-2x
dv/dx = -1 + 2x
dv/dy = -2y
This means -2y = -2y and 1-2x = 1-2x and f(z) is differentiable everywhere.
So its differentiable at points z(x,y) and its derivative is df/dx = -2y + i(-1 + 2x) at all points.
Can someone just check that this work is accurate and stated correctly? I'm just learning this material.
Thanks in advance for your help.