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 the real part of approaches ; going away from the real axis into the upper half of the gives function that also approaches as . Along the real axis the imaginary part of vanishes identically; going away from the real axis into the gives a function that approaches as . 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 real part has discontinuities along the branch cuts. The viewpoint is from the left half‐plane. At the function has logarithmic singularities. The real part and the imaginary part of over the . The left graphic shows and the right graphic shows . The viewpoint is from the left half‐plane. No branch point exists at , just two disconnected sheets. 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 left and right neighborhood of the branch cuts. Only the real part shows discontinuities due to the branch cuts. The viewpoint is from the left half plane. The rectangular shape of the vertical walls arise from the logarithmic singularities at . 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 viewpoint is from the left half‐plane. No branch point exists at , just two disconnected sheets. 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 ; going away from the real axis into the upper half of the gives function that also approaches as . The real part is discontinuous along the branch cuts , . 
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 ; going away from the real axis into the upper half of the gives a function that also approaches as . 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; going away from the real axis into the gives function that approaches as . Along the imaginary axis has a branch cut on the intervals and . is a continuous, but not continuously differentiable, function over the . The logarithmic singularities at are 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 as . Along the imaginary axis, has a branch cut on the intervals and . is a continuous, but not continuously differentiable, function over the . The logarithmic singularities at are 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. The logarithmic singularities at are 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 functions value 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. 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. as spikes extending upwards. The zero at and the logarithmic singularity 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. At the function has no singularity and two disconnected sheets are 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. At the function has no singularity and two disconnected sheets are visible. 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 no singularity. 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 no singularity. 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 no singularity. 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 no singularity. 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. 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. as spikes extending upwards. At the function has no singularity. At a zero from a logarithmic branch point exists.
