Answer to Question #204302 in Complex Analysis for Frank

Question

Interpret each of the following transformation in the complex plane

(i) T1:z→ w, given by w=-z* where z* is the conjugate of z.

(ii) T2:z→w given by w=3z-1+2i.

Find the invariant complex number under the transformation T2.

Answer 1

Let us interpret each of the following transformation in the complex plane.

(i) $T_1:z→ w$, given by $w=-z^*$ where $z^*$ is the conjugate of $z$. Let $z=a+ib,$ where $a,b\in\R.$ Then $z^*=a-ib$ and $-z^*=-a+ib.$ Therefore, $T_1: a+ib\to -a+ib.$ It follows that $T_1$ is a reflection in the imaginary axis.

(ii) $T_2:z→w$ given by $w=3z-1+2i$. It follows that the map $w\mapsto 3w$ is dilation with scale factor $3$ and with the center in origin, and $z\mapsto z-1+2i$ is a traslation of 1 unit to the left and 2 units upwards. Therefore, $T_2:z→w$ is a composition of this two transformations.

Let us find the invariant complex number under the transformation $T_2$:

$z=3z-1+2i$

$2z=1-2i$

$z=0.5-i$

Consequently, $0.5-i$ is the invariant complex number under the transformation $T_2.$