The Complex Sine Function
Drag the blue points to see the effect of taking the sine of a complex number/point for points on various shapes. The checkboxes show different shapes. The "before" shape is filled in, and is traced by the blue point P. The "after" shape is not filled, and is traced by P'.
The complex sine function can be defined as
sin(x+iy) = sin(x) cosh(y) + i cos(x) sinh(y))
(cosh and sinh are the hyperbolic cosine and sine functions, respectively.)