3D plots over the complex plane

Entering the complex plane

in the near the real axis. Here the complex variable is expressed as . The red surface is the real part of . The blue, semitransparent surface is the imaginary part of . The pink tube is the function along the real axis. Along the real axis the real part of is oscillating and has first‐order poles at . Going away from the real axis into the gives a function that approaches 0 exponentially. Along the real axis, the imaginary part of vanishes identically; going away from the real axis into the gives a function that approaches exponentially.

Real part over the complex plane

The real part of where . The surface is colored according to the imaginary part. The right graphic is a contour plot of the scaled real part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. Along the real axis, the real part of is oscillating; going away from the real axis into the gives a function that approaches 0 exponentially. The first‐order poles at are clearly visible.

The absolute value of the real part of where . The surface is colored according to the absolute value of the imaginary part. The right graphic is a contour plot of the scaled absolute value of the real part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. Along the real axis, the real part of is oscillating; going away from the real axis into the gives a function whose absolute value of the real part approaches 0 exponentially. The first‐order poles at are clearly visible.

Imaginary part over the complex plane

The imaginary part of where . The surface is colored according to the real part. The right graphic is a contour plot of the scaled imaginary part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. Along the real axis, the imaginary part of vanishes identically; going away from the real axis into the gives a function that approaches exponentially. The first‐order poles at are clearly visible.

The absolute value of the imaginary part of where . The surface is colored according to the absolute value of the real part. The right graphic is a contour plot of the scaled absolute value of the imaginary part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. Along the real axis, the imaginary part of vanishes identically; going away from the real axis into the gives a function that approaches exponentially. The first‐order poles at are clearly visible.

Absolute value part over the complex plane

The absolute value of where . The surface is colored according to the argument. The right graphic is a contour plot of the scaled absolute value, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. Along the real axis has poles; going away from the real axis gives a function that approaches 0 exponentially. The first order poles at are clearly visible.

Argument over the complex plane

The argument of where . The surface is colored according to the absolute value. The right graphic is a contour plot of the scaled argument, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. has lines of discontinuities over the .

The square of the sine of the argument of where . For dominantly real values the function values are near 0, and for dominantly imaginary values, the function values are near 1. The surface is colored according to the absolute value. The right graphic is a cyclically colored contour plot of the argument. Red represents arguments near and light‐blue represents arguments near 0.

Zero-pole plot

The logarithm of the absolute value of where in the upper half‐plane. The surface is colored according to the square of the argument. In this plot zeros are easily visible as spikes extending downwards and poles and logarithmic singularities as spikes extending upwards.

Real part over the complex plane near infinity

The real part of where . The surface is colored according to the imaginary part. The right graphic is a contour plot of the scaled real part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. The essential singularity at results in a complicated structure that cannot be resolved graphically. The accumulation of poles near cannot be resolved graphically.

The absolute value of the real part of where . The surface is colored according to the absolute value of the imaginary part. The right graphic is a contour plot of the scaled absolute value of the real part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. The essential singularity at results in a complicated structure that cannot be resolved graphically. The accumulation of poles near cannot be resolved graphically.

Imaginary part over the complex plane near infinity

The imaginary part of where . The surface is colored according to the real part. The right graphic is a contour plot of the scaled imaginary part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. The essential singularity at results in a complicated structure that cannot be resolved graphically. The accumulation of poles near cannot be resolved graphically.

The absolute value of the imaginary part of where . The surface is colored according to the absolute value of the real part. The right graphic is a contour plot of the scaled absolute value of the imaginary part, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. The essential singularity at results in a complicated structure that cannot be resolved graphically. The accumulation of poles near cannot be resolved graphically.

Absolute value part over the complex plane near infinity

The absolute value of where . The surface is colored according to the argument. The right graphic is a contour plot of the scaled absolute value, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. The essential singularity at results in a complicated structure that cannot be resolved graphically. The accumulation of poles near cannot be resolved graphically.

Argument over the complex plane near infinity

The argument of where . The surface is colored according to the absolute value. The right graphic is a contour plot of the scaled argument, meaning the height values of the left graphic translate into color values in the right graphic. Red is smallest and violet is largest. The essential singularity at results in a complicated structure that cannot be resolved graphically. The accumulation of poles near cannot be resolved graphically.

The square of the sine of the argument of where . For dominantly real values the function values are near 0, and for dominantly imaginary values, the function values are near 1. The surface is colored according to the absolute value. The right graphic is a cyclically colored contour plot of the argument. Red represents arguments near and light‐blue represents arguments near 0.

Zero-pole plot near infinity

The logarithm of the absolute value of where in the upper half‐plane. The surface is colored according to the square of the argument. In this plot zeros are easily visible as spikes extending downwards and poles and logarithmic singularities as spikes extending upwards. The essential singularity at results in a complicated structure. The accumulation of poles near cannot be resolved graphically.