What is the effect of the transformation w = az + b when a is a complex number and b is zero?

The transformation w = az when a is a complex number and b is zero results in a rotation and magnification of the z-plane. Specifically, the magnitude of the transformation is determined by |a|, and the angle of rotation is given by the argument of a.

When applying the transformation w = az with a being a complex number and b being zero, the effect is twofold: it magnifies distances by a factor of |a| and rotates points in the z-plane by an angle equal to the argument of a. This means that the original straight lines in the z-plane will still map to straight lines in the w-plane, but their orientation and lengths will change according to the properties of the complex number a.