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:
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $71 \ \text {(e)}$