Complex Modulus/Examples/3iz - z^2

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Example of Complex Modulus

Let:

$w = 3 i z - z^2$

where $z = x + i y$.

Then:

$\cmod w^2 = x^4 + y^4 + 2 x^2 y^2 - 6 x^2 y - 6 y^3 + 9 x^2 + 9 y^2$


Proof

\(\displaystyle \cmod w^2\) \(=\) \(\displaystyle \cmod {3 i z - z^2}^2\) Definition of $w$
\(\displaystyle \) \(=\) \(\displaystyle \cmod {3 i \paren {x + i y} - \paren {x + i y}^2}^2\) Definition of $z$
\(\displaystyle \) \(=\) \(\displaystyle \cmod {3 i x - 3 y - x^2 + y^2 - 2 i x y}^2\) multiplying out
\(\displaystyle \) \(=\) \(\displaystyle \paren {-x^2 + y^2 - 3 y}^2 + \paren {3 x - 2 x y}^2\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle \paren {x^4 - 2 x^2 y^2 + 6 x^2 y + y^4 - 6 y^3 + 9 y^2} + \paren {9 x^2 - 12 x^2 y + 4 x^2 y^2}\)
\(\displaystyle \) \(=\) \(\displaystyle x^4 + y^4 - 2 x^2 y^2 + 4 x^2 y^2 + 6 x^2 y - 12 x^2 y - 6 y^3 + 9 x^2 + 9 y^2\)
\(\displaystyle \) \(=\) \(\displaystyle x^4 + y^4 + 2 x^2 y^2 - 6 x^2 y - 6 y^3 + 9 x^2 + 9 y^2\)

$\blacksquare$


Sources