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Consider a complex number in rectangular form, i.e. $z=x+iy$. If I am given the real part and imaginary part of the function, take $z^2$ as an example, i.e. $f(z)=z^2=(x^2-y^2)+i(2xy)$. Is it possible for me to have a systematic way to find out the original function? (i.e. $f(z)=z^2$)

I don't know whether it is a silly question, or the answer can be found easily from google. So any related link or generous response will mean a lot to me!

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  • $\begingroup$ Just to clarify, you're supposing we know both the real and imaginary parts? $\endgroup$ – πr8 Sep 26 '16 at 9:30
  • $\begingroup$ yes. Both of them are known $\endgroup$ – tomchan516 Sep 26 '16 at 10:00
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You can substitute $$x=\frac{z+\bar z}{2},\quad y=\frac{z-\bar z}{2\,i}.$$ If everything is alright, the terms in $\bar z$ should disappear.

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