How do you illustrate the image of a point in the complex z-plane under a complex mapping onto the w-plane?

To illustrate the image of a point in the complex z-plane under a complex mapping onto the w-plane, you plot the values of z on the z-plane and the corresponding values of w=f(z) on the w-plane. This involves using the transformation equation w = f(z) = u(x, y) + v(x, y) to determine the new coordinates in the w-plane.

In the complex z-plane, each point z can be represented as z = x + jy, where x and y are real numbers. The mapping w = f(z) transforms these points into the w-plane, where w is expressed as w = u(x, y) + jv(x, y). Since this transformation involves four real variables (x, y, u, and v), you cannot plot them on a single set of axes. Instead, you plot the z-values on the z-plane and their corresponding w-values on the w-plane, effectively visualizing how points in the z-plane map to points in the w-plane based on the transformation defined by f(z).