Complex Modulus/Examples/3iz - z^2

Example of Complex Modulus

Let:

$w = 3 i z - z^2$

where $z = x + i y$.

Then:

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

$\ds \cmod w^2$

$=$

$\ds \cmod {3 i z - z^2}^2$

Definition of $w$

$\ds $

$=$

$\ds \cmod {3 i \paren {x + i y} - \paren {x + i y}^2}^2$

Definition of $z$

$\ds $

$=$

$\ds \cmod {3 i x - 3 y - x^2 + y^2 - 2 i x y}^2$

multiplying out

$\ds $

$=$

$\ds \paren {-x^2 + y^2 - 3 y}^2 + \paren {3 x - 2 x y}^2$

$\ds $

$=$

$\ds \paren {x^4 - 2 x^2 y^2 + 6 x^2 y + y^4 - 6 y^3 + 9 y^2} + \paren {9 x^2 - 12 x^2 y + 4 x^2 y^2}$

$\ds $

$=$

$\ds x^4 + y^4 - 2 x^2 y^2 + 4 x^2 y^2 + 6 x^2 y - 12 x^2 y - 6 y^3 + 9 x^2 + 9 y^2$

$\ds $

$=$

$\ds x^4 + y^4 + 2 x^2 y^2 - 6 x^2 y - 6 y^3 + 9 x^2 + 9 y^2$

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $60$