Complex tangent

tan z = (sin x cos x + i sinh y cosh y) / (cos2 x + sinh2 y)

For real numbers (sinh y =0 ) the tangent function diverges periodically where

cos x =0 , while the sign changes. To observe this sensitive behavior in the neigborhood of a divergence one should input the coordinates of the array edge point as numbers in the x, y fields

With real numbers (y = 0; sinh y = 0; cosh y = 1) points at the real axis are mapped periodically with period 2π into -∞ ≤ tan x ≦ ∞. Strips of the real axis of the z-plane of width 2π are mapped to -∞ ≤ x ≤ ∞ in the u-plane, hence in a full Riemann sheet.

Points on parallels to the real axis are mapped to closed curves around +i and -i. Points on parallels to the imaginary axis are mapped to curves ending at +i and -i. Points with imaginary value larger than π are mapped into point i, points with imaginary value smaller than -π are mapped into point -i.

Points with finite imaginary part are mapped periodically to finite areas. Points with no imaginary part are mapped periodically to -∞ ≤ x ≤ +∞.

Four red lines show the period limits in x direction, and the mapping limits in y direction.

Shifting the quadratic array along the real axis by rotates the image once around +i, -i. If the array has only positive or negative imaginary parts, the image will rotate around one of these poles, and there will be no divergence.

The circular array around the origin with radius π/4 is mapped to an oval with the accentuated points at (-1,0) and (+1,0). Shifting along the real axis both points move with different velocity. When the first one passes the divergence, the sign changes and the mapped pattern changes to its mirror image. When the circle in the z-plane contains no real values, it image rotates around -i or i with no divergence.