I realize this is probably a silly question but I'm going to have to ask it anyways: how would I define a complex function so that is a perfect horizontal line, perpendicular to the $iy$ axis and of course never touching the $x$ axis?
Let's let $y=c$ where $c$ is a non-zero constant. Then is my function $z=0x+ic=ic$, similar to how we might make a horizontal function between two real axes? Or is it $z=x+ic$?
Sorry, having a bad math day. Any help would be greatly appreciated.
You can parametrize the line as $z=x+ic \mid x \in \mathbb{R}\,$, or write its complex equation as $z-\bar z=2ic\,$.