A real function f(x) is a curve. A complex function f(z) maps a 2D plane to another 2D plane — so we need new tricks to visualize it. Here are the two main ones we'll use all course: domain coloring and grid deformation.
Color = argument (angle) of f(z), brightness = how close |f(z)| is to a whole number (shows "height" contours). Hover to inspect.
A square grid in the input plane (left domain) gets bent into curves by f. Notice: the grid lines still cross at right angles — that's "conformality," a hallmark of complex differentiability.