Linear and Quadratic function on $\mathbb{C}$

1. Linear function on $\mathbb{C}$

$f(z) = \alpha z + \beta$     $(\alpha, \beta \in \mathbb{C})$

         $= \alpha (z + \frac{\beta}{\alpha})$

         $= \frac{\alpha}{\left|\alpha \right| } \cdot \left|\alpha \right| (z + \frac{\beta}{\alpha})$

 

$+ \frac{\beta}{\alpha}$  :  translate

$\left|\alpha \right| $  :  times

$\frac{\alpha}{\left|\alpha \right| }$  :  CCW rotation

 

 

 

Ex

$w = (1+i)z + 2$

Since $1 + i = \sqrt{2} (\cos \frac{\pi}{4} + \sin \frac{\pi}{4})$,

$\Rightarrow$  $\sqrt{2}$ times + $\frac{\pi}{4}$ CCW rotation + translated by $(2, 0)$

 

 

 

2. Quadratic function on $\mathbb{C}$

$f(z) = \alpha z^{2} + \beta z + \gamma$     $(\alpha, \beta, \gamma \in \mathbb{C})$

         $= \alpha(z + \frac{\beta}{2 \alpha})^{2} + \gamma_{1} - \frac{\beta^{2}}{4 \alpha}$

 

$g(z) = z + \frac{\beta}{2 \alpha}$  :  translate

$h(z) = z^{2}$

$k(z) = \alpha z + \gamma - \frac{\beta^{2}}{4 \alpha}$  :  linear function

 

$\therefore$  $f(z) = k \circ h \circ g(z)$

 

 

 

2-1. $z^{2}$ on $\mathbb{C}$

$f(z) = z^{2}$

$z = x + iy \mapsto z^{2} = (x^{2} - y^{2}) + i(2xy)$

 

$F : \mathbb{R}^{2} \to \mathbb{R}^{2}$ by

$(x, y) \mapsto (x^{2} - y^{2}, 2xy)$

 

Let $u = x^{2} - y^{2}$,  $v = 2xy$

 

$\Rightarrow$

$x^{2} = \dfrac{\sqrt{u^{2} + v^{2}} + u}{2} $

$y^{2} = \dfrac{\sqrt{u^{2} + v^{2}} - u}{2} $

 

 

 

Ex 1  $y = mx$

$z^{2} : (x, mx) \mapsto (x^{2}(1 - m^{2}), 2mx^{2})$

 

$\left\{\begin{matrix}
u = x^{2}(1-m^{2}) \\
v = 2mx^{2} \end{matrix}\right.$     $\Rightarrow$     $v = \dfrac{2m}{1-m^{2}}u$

 

$\mathrm{arg}(z) = \theta$ or $\mathrm{arg}(z) = \theta + \pi$     $\Rightarrow$     $\mathrm{arg}(z^{2}) = 2\theta$

 

 

 

Ex 2  $y = c$

$z^{2} : (x, c) \mapsto (x^{2} - c^{2}, 2cx)$

 

$\left\{\begin{matrix}
u = x^{2} - c^{2} \\
v = 2cx \end{matrix}\right.$     $\Rightarrow$     $u = \dfrac{v^{2}}{4c^{2}} - c^{2}$

 

 

 

Ex 3  $x = c$

$z^{2} : (c, y) \mapsto (c^{2} - y^{2}, 2cy)$

 

$\left\{\begin{matrix}
u = c^{2} - y^{2} \\
v = 2cy \end{matrix}\right.$     $\Rightarrow$     $u = -\dfrac{v^{2}}{4c^{2}} + c^{2}$

 

 

 

Ex 4

$\left\{z \; | \; 0 \leq \mathrm{arg}(z) \leq \frac{\pi}{2} \right\}$     $\Rightarrow$     $\left\{z \; | \; 0 \leq \mathrm{arg}(z^{2}) \leq \pi \right\}$

 

 

 

Ex 5

$\left\{z \; | \; \left|z \right| \leq 1 \right\}$     $\Rightarrow$     $\left\{z \; | \; \left|z \right| \leq 1 \right\}$