TFunctionQuadrupleCovering - crowlogic/arb4j GitHub Wiki

• The function $f(z) = z^2$ maps the entire complex plane to the right half-plane (excluding the negative real axis), since squaring a complex number squares its magnitude and doubles its argument. Thus, every point in the right half-plane has two pre-images, one in each of the two halves of the plane.

• The function $g(z) = \ln(1+z)$ further maps the right half-plane to the entire complex plane (excluding the negative real axis), duplicating each point. Thus, every point in the entire complex plane has two pre-images in the right half-plane.

• Finally, the function $h(z) = \tanh(z)$ maps vertical stripes of width πi in the complex plane onto the entire complex plane.

In this sense, the composite map $f(z) = \tanh(\ln(1+z^2))$ acts as a quadruple cover over the complex plane, given that every point in the complex plane has four pre-images, one in each quadrant of the pre-image plane.