# Thread: Curve Sketching: Complex domain

1. ## Curve Sketching: Complex domain

Let $f(z)=z^2$
Find the images of the line $Re(z)=0$ and $Im(z)=1$ under the above mapping, $w=f(z)$

Just to make sure I'm understanding this curve sketching
Represented parametrically, $Re(z)=0$ is $z_1(t)=it$ and $Im(z)=1$ is $z_2(t)=t+i, t\in{\mathbb{R}}$

So $f(z_1(t))=-t^2+0i$ and $f(z_2(t))=t^2-1 +2ti$
$x_1=-t^2, x_2=t^2-1$
$y_1=0, y_2=2t$

$x_2=\frac{y^2}{4}-1$

Thus the line $Re(z)=0$ is mapped onto the negative portion of the x-axis and the line $Im(z)=1$ is mapped onto this parabola: $x_2=\frac{y^2}{4}-1$
Any mistakes present?