Complex Sine Function
The function is w = sin(z). The red grid is in the
z = x+iy domain, the blue image is in the
w = u+iv domain.
Along lines parallel to the real axis, sin behaves similarly to its real counterpart
and is periodic.
However, along paths parallel to the imaginary axis, sin appears to be
decaying and then growing. This result is initially surprising, but becomes clear when
w = sin(z) = (eiz - e-iz) / (2i)
is expanded to become
2i sin(x+iy)
= e-y eix
- ey e-ix
so that
2i sin(x+iy)
= e-y [cos(x) + i sin(x]
- ey [cos(-x) + i sin(-x)]
We see that the respective terms grow as y takes on large negative/positive
values.