Complex Power Functions in Polar Coordinates

Rather than using a square domain

, we can restrict the domain in a different way. In the case of a real variable, we used as domain -1 < x < 1, which can also be interpreted as the set of x with |x| < 1, where |x| stands for the absolute value of x. Then |xⁿ| = |x|ⁿ < 1 as well, so the graphs stay within the unit square.

For a complex number z = x + yi, we also have the idea of an absolute value, or norm, given by |z| = (x² + y²)^1/2, which is the length of the vector (in the plane) with coordinates (x,y). If we restrict the domain of the nth-power function, w = zⁿ, to |z| < 1, then we have |w| = |zⁿ| = |z|ⁿ < 1 so the graph stays within the hypercube in four-space.

The complex identity function

The complex squaring function

The complex cubing function

		The complex identity function
		The complex squaring function
		The complex cubing function