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, semi‐transparent 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 real part of is a continuous function with singularities at ; going away from the real axis into the gives a function that approaches as . Along the real axis, the imaginary part of is piecewise constant; going away from the real axis into the is a function that approaches . At has logarithmic singularities. In the interval the function has upper lip continuity, in the interval the function has lower lip continuity. 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 logarithmic singularities. 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 regular point of . Two disconnected sheets exist there. 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 neighborhoods of the branch cuts. The branch points at are logarithmic singularities. Only the imaginary part shows discontinuities due to the branch cuts. The viewpoint is from the lower 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 neighborhoods of the branch cuts. is a regular point of . Two disconnected sheets exist there. 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 approaches 0 as ; going away from the real axis into the also gives a function that approaches 0. 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 approaches 0 as ; going away from the real axis into the also gives a function that approaches 0. 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 has branch cuts in the intervals and ; going away from the real axis into the gives function that approaches . is a discontinuous function over the along the branch cuts and . The branch points at are logarithmic branch points. 
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. As , the absolute value of the imaginary part 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 tangent 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 cyclical colored contour plot of the argument. Red represents arguments near and light‐blue represents arguments near 0. Zeropole 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 zero at and the logarithmic singularities at are 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. is a regular point of the 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. 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. is a regular point of the function . 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. is a regular point of the function . 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. is a regular point of the function . The square of the tangent of the argument of where . For dominantly real values, the 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 cyclicaly colored contour plot of the argument. Red represents arguments near and light‐blue represents arguments near 0. Zeropole 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.. is a regular point of the function .
