Given the set $A=\{z\in\mathbb{C}:\lvert z\rvert >5, \Re(z)<0\}$, how to represent $B=\{w\in \mathbb{C}: w= -iz+2, z\in A\} $?

Question

Of course $A$ is the left semiplane deprived of the semicircle of radius $5$. I haven't been able to make $B$ explicit writing $w$ in trigonometric notation, and I don't think that is the way. Using the algebraic notation, $B=\{w\in \mathbb{C}: w= y+2-ix, (x,y)\in A\}.$ I thought I should make some transformations to the graph of $A$: firstly I would shift it up by two units ($y\mapsto y+2$), then reflect the result about the y-axis ($x\mapsto -x$) and then swap the new x and y coordinates (which should mean rotate the graph by 90 degrees clockwise and then rotate it by 180 degrees around the new x-axis). Is this correct?

Answer 1

Yes , though a more effective way of describing the last step is "reflect the graph about the line y=x "