The Complex Function θ and Related Graphs

The scheme for these graphs is that a color is attached to each point in the plane, say color(z), and a function w = f(z) is graphed by attaching color(w) to the pixel at location z. The coloring scheme is that a color is associated to each sector of angle π/4 radiating from the origin, and the colors are darkened in the ring between radii 1 and e, e² and e³, ..., e²ⁿ and e²ⁿ⁺¹, ... To illustrate, here is the graph of the identity function. All graphs have corners at ± 80/3 ± 20 i , so the fact that the largest ring is nearly tangent to the top and bottom of the graph follows from the fact that e³ is approximately 20 :

Here is the graph of θ as a complex function, along with θ − π/2, which, because θ is symmetric about π/2, is an odd function:

θ                                                                                             θ − π/2

θ oscillates up and down around the value 0 on the negative real axis. The three points surrounded by the full cycle of eight colors are the zeros. The points on the real axis where a pair of yellow areas meet a pair of green areas (or pink and blue) represent local maxima or minima. There are several radial lines where there are abrupt changes in coloring. These represent cuts generated by ramification around which the function is multi-valued. There are two such points visible in each quadrant, as well as a pair of less obvious ones on the imaginary axis.

The change in value of θ around each singularity turns out to be 4π, so cos θ, which is actually the function that determines the metric for the hyperbolic square, is single-valued. Also, tan (θ − π/2)/4 is single valued. The points where the colors cycle around clockwise, rather than counter-clockwise, and have target-like rings around them, are poles. Those in the cosine graph are double poles, those in the tangent are single.

cosθ tan (θ − π/2)/4