FOlks,

Use the C-R equations to determine whether f(z)=u(x,y)+iv(x,y) is complex analytic for the following, if so calculate f'(z).

a) $\displaystyle \frac{x-iy}{x^2+y^2}$

1) Is complex analytic the same as complex differentiable?

2) I determine f(z) = $\displaystyle \frac{x}{x^2+y^2}$ +i $\displaystyle \frac{-y}{x^2+y^2}$ and $\displaystyle f'(z) =u_x+i v_x$

Using the quotient rule I determine $\displaystyle u_x=v_y= \frac{-x^2+y^2}{(x^2+y^2)^2}$ and $\displaystyle u_y=-v_x= \frac{-2xy}{(x^2+y^2)^2}$

Therefore $\displaystyle f'(z)= \frac{-x^2+y^2}{(x^2+y^2)^2}+ i \frac{2xy}{(x^2+y^2)^2}$...?