Question 2.44: Find a mapping function which maps the points z = 0, ±i, ±2i...

Since the points in the z plane are equally spaced, we are led, because of Problem 2.15, to consider a logarithmic function of the type z = ln w.

Now, if w=1=e^{2 k \pi i}, k=0, ±1, ±2, … , then z=ln w=2kπi so that the point w=1 is mapped into the points 0, ±2πi, ±4πi, …

If, however, we consider z =(ln w)/2π, the point w = 1 is mapped into z = 0, ±i, ±2i, … as required. Conversely, by means of this mapping function, the points z = 0, ±i, ±2i, … are mapped into the point w=1.

Then, a suitable mapping function is z = (ln w)/2π or w=e^{2 \pi z} .