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Can someone please answer me, HOW does $Im(z^2) = 4$ get graphed like this?

enter image description here

and not like a normal parabola?

Like $Re(z^2) = 4$

enter image description here

But of course on the Imaginary axis.

It has been eating my mind up - and I just can't explain why the graph does that. Help would be greatly appreciated.

F.A.
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MATHSUSER
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1 Answers1

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Look at this way and see if you're satisfied. If we let $z=x+iy$ so $$\text{Im}(z^2)=2xy,~~\text{Re}(z^2)=x^2-y^2$$ so you can simply plot $xy=2$ when $\text{Im}(z^2)=4$. The following plot is a conformal plot of $\text{Im}(z^2)-4$ in an interval:

enter image description here

Mikasa
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  • but if xy = 2 , it gives you a completely new graph. It turns in to a hyperbola. – MATHSUSER Nov 21 '13 at 10:30
  • So are you saying WolframAlpha has screwed up its graphing for complex numbers, because it gave me the right answer for Real. – MATHSUSER Nov 21 '13 at 10:33
  • @MATHSUSER: I don't know Mathematica but the theoretical approach is correct. Maybe we should wait to get another point of views. ;-) – Mikasa Nov 21 '13 at 10:40
  • Alright, thanks though considering no one else is answering. But I think WolframAlpha has made a mistake with Its complex number graphing. – MATHSUSER Nov 21 '13 at 10:41