$\displaystyle f(z)=tan^{-1}(\frac{2xy}{x^{2}-y^{2}})-iln(|z|^{2}) $ how can we write f(z) in terms of z.
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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 $\displaystyle \log w = \log|w| + i\arg w$.
mm. i got : $\displaystyle arg(z^{2}) - i log (|z|^{2}) $ can it be any simpler?
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.
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