FunctionFamilies.html

Complex Sine Function

The function is w = sin(z). The blue grid is in the z = x+iy domain, the red 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) = (e^iz - e^-iz) / (2i) is expanded to become 2i sin(x+iy) = e^-y e^ix - e^y e^-ix
so that
2i sin(x+iy) = e^-y [cos(x) + i sin(x] - e^y [cos(-x) + i sin(-x)] We see that the respective terms grow as y takes on large negative/positive values.