Complex Hyperbolic Tangent Function

tanh 1domain to final
function: z → tanh(z/2) = sinh(z/2)/cosh(z/2) = (exp(z) - 1)/(exp(z) + 1)
domain: -1.7 ≦ Re(z) ≦ 1.7, -pi ≦ Im(z) ≦ pi.
Since i*tan(z) = tanh(i*z), one can use this image to visualize the complex tangent function, just rotate the image through 90 degrees.
tanh 2domain to polar
It is useful to visualize this function in 2 steps: z → w = exp(z) → (w - 1)/(w + 1). This is the 1st step, the exponential map from the cartesian grid to the conformal polar grid.
tanh 3polar to final
This is the second step w → (w - 1)/(w + 1), from the polar grid to the tanh-image,

z_hyperbolic_tangent.pdf