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. At , the function has logarithmic singularities. Along the real axis outside the interval , the imaginary part of vanishes identically; going away from the real axis into the gives a function that approaches 0. Along the real axis, the imaginary part of is piecewise constant; going away from the real axis into the is a function that approaches . The imaginary part is discontinuous along the branch cuts . The imaginary part has lower lip continuity in the interval and upper lip continuity in the interval . 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. At , the function has logarithmic branch points. The real part and the imaginary part of over the . The left graphic shows and the right graphic shows . The viewpoint is from the lower half‐plane. is a regular point of . 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 branch points. Only the imaginary part shows discontinuities due to the branch cuts. 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. is a regular point of . The viewpoint is from the lower 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. The function has logarithmic singularities at ; going away from the real axis into the upper half of the gives a function that asymptotically 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. The function has logarithmic singularities at ; going away from the real axis into the upper half of the gives a function that asymptotically 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 is piecewise constant; going away from the real axis into the gives a function that approaches . is a discontinuous function over the along the interval . 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. Along the real axis, the imaginary part of is piecewise constant; going away from the real axis into the gives function that approaches . is a discontinuous function over the along the interval . The branch points at are logarithmic branch points. 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 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. 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 logarithmic branch points 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. 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. 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. 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. 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 square of the sine 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 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.
