1. ## complex analysis

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?

2. Hello !
Originally Posted by Kai
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?
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 =)

3. Originally Posted by Moo
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 =)
Yep, easy in the end, my mind not working, maybe exam stress