# Thread: how can write this expression in z=x+yi

1. ## how can write this expression in z=x+yi

$\displaystyle f(z)=tan^{-1}(\frac{2xy}{x^{2}-y^{2}})-iln(|z|^{2})$

how can we write f(z) in terms of z.

2. Hints. (1) If z = x + iy then $\displaystyle z^2 = x^2-y^2 + 2xyi$.

(2) If w = u + iv then $\displaystyle \arg w = \tan^{-1}\bigl(\tfrac vu\bigr) \pm \pi$.

(3) If w = u + iv then $\displaystyle \log w = \log|w| + i\arg w$.

3. ## mm

mm. i got :

$\displaystyle arg(z^{2}) - i log (|z|^{2})$

can it be any simpler?

4. Originally Posted by szpengchao
mm. i got :

$\displaystyle arg(z^{2}) - i log (|z|^{2})$

can it be any simpler?
Yes. That was the point of hint 3.