3D plots over the complex plane

Entering the complex plane

Upper picture: in the upper half of the near the real axis viewed from the lower half‐plane. Lower picture: in the lower half of the near the real axis viewed from the upper half‐plane. 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 real part of the function along the real axis and the skyblue tube is the imaginary part of the function along the real axis. Along the real axis inside the interval , the real part of is piecewise constant; going away from the real axis into the gives a logarithmically increasing function. Along the real axis, the imaginary part of vanishes identically outside the interval ; going away from the real axis into the is a function that approaches . The imaginary part is discontinuous along the branch cuts and . The imaginary part has upper lip continuity in the .

Branch cuts

The real part and the imaginary part of over the . The left graphic shows and the right graphic shows . Along the intervals , the function has branch cuts. The imaginary part has discontinuities along the branch cuts. The branch points at are of square root type. The viewpoint is from the lower half‐plane.

The real part and the imaginary part of over the . The left graphic shows and the right graphic shows . is a logarithmic branch point of . The viewpoint is from the upper half‐plane.

The branch cuts of the real part and the imaginary part of over the . The left graphic shows and the right graphic shows . The red and blue vertical surfaces connect points from the immediate lower and upper neighborhood of the branch cuts. The branch points at are of square root type. Only the imaginary part shows discontinuities due to the branch cuts. The viewpoint is from the lower half‐plane. At , two branch cuts start to coincide, but only one branch point is there.

The branch cuts of the real part and the imaginary part of over the . The left graphic shows and the right graphic shows . The red and blue vertical surfaces connect points from the immediate lower and upper neighborhoods of the branch cuts. The logarithmic singularity at and the corresponding branch cut are visible. Only the imaginary part shows discontinuities due to the branch cuts. The viewpoint is from the upper half‐plane.

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 a logarithmically increasing function; going away from the real axis into the upper half of the gives a logarithmically increasing function.

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 logarithmically increasing function; going away from the real axis into the upper half of the gives a function that with increasing imaginary part approaches . The real part is discontinuous along the branch cuts , .

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 outside the branch cut intervals , ; going away from the real axis into the gives a function that approaches . is a discontinuous function over the . The branch points at are of square root type.

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 outside the branch cut intervals , ; going away from the real axis into the gives a function that approaches . The branch points at are of square root type.

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.

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. The square root branch point at is visible.

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. At , the function has a logarithmic branch point.

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.

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. At , the function has a logarithmic branch point.

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. At the function has a logarithmic branch point.

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. At , the function has a logarithmic branch point.

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. At , the function has a logarithmic branch point.

The square of the sine of the argument of where . For dominantly real functions 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. At , the function has a logarithmic singularity.