Plate 4

Visualizations of complex-valued functions. The domain zeta = x + i y is represented in the x-y plane. The height of the surface (z-axis) is equal to the absolute value |f(zeta)|. The argument or phase angle arg(f(zeta)) is color-coded by hue. Red is positive real. Upper left: sin(zeta), upper right: e^zeta. Lower left: e^{1/zeta}, which has an essential singularity at 0. Lower right: 1/zeta^2, which has a double pole at 0.