Equation of Hyperbola in Complex Plane/Examples/Imaginary Part of z^2 = 4

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Example of Use of Equation of Hyperbola in Complex Plane

The equation:

$\map \Im {z^2} = 4$

describes a hyperbola embedded in the complex plane.


Proof

\(\ds \map \Im {z^2}\) \(=\) \(\ds 4\)
\(\ds \leadsto \ \ \) \(\ds \map \Im {\paren {x + i y}^2}\) \(=\) \(\ds 4\)
\(\ds \leadsto \ \ \) \(\ds \map \Im {x^2 - y^2 + 2 i x y}\) \(=\) \(\ds 4\)
\(\ds \leadsto \ \ \) \(\ds 2 x y\) \(=\) \(\ds 4\)
\(\ds \leadsto \ \ \) \(\ds x y\) \(=\) \(\ds 2\)

This is the equation of a rectangular hyperbola:


Equation of Hyperbola in Complex Plane-Examples-Im x^2 = 4.png


$\blacksquare$


Sources