Complex Exponential Function

The function is w = exp(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, exp behaves like its real counterpart.


However, along paths parallel to the imaginary axis, exp is shown to be a periodic function. This result is initially surprising, but becomes obvious considering w = ex+iy written as
w = ex eiy = ex(cos(y) + i sin(y)).