let z = x + iy and let
f(z) = $\displaystyle 2x(y+1) + i(x^2+2y-y^2)$
find f(z) in terms of z and $\displaystyle \bar z$
i only need a start-up
anybody help?
Hello !
If $\displaystyle z=x+iy$, $\displaystyle \bar z=x-iy$
Note that $\displaystyle z+\bar z=2x$ and $\displaystyle z-\bar z=2iy$
Hence we have :
$\displaystyle \boxed{x=\frac{z+\bar z}{2}} \quad \quad \boxed{y=\frac{z-\bar z}{2i}}$
It may be all you need here
Edit : notice that if you still have $\displaystyle \bar z$ in the expression of f(z), then the function is not holomorphic =)