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 outside the interval , the real part of is constant; going away from the real axis into the gives a function that approaches . Along the real axis, the imaginary part of vanishes identically in the interval going away from the real axis into the is a function that approaches logarithmically. The real part is a continuous function, but not a continuously differentiable function, and the imaginary part is discontinuous along the branch cuts and . The imaginary part has upper lip continuity in the left half‐plane and lower lip continuity in the right half‐plane. 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 upper half‐plane. The real part and the imaginary part of over the . The left graphic shows and the right graphic shows . The logarithmic singularity at and the corresponding branch cut are visible. 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. 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 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 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 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. is a continuous function, but not everywhere continuously differentiable function over the . Along the real axis outside the interval , the real part of is constant.; going away from the real axis into gives a function that approaches as . 
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. is a continuous function, but not a everywhere continuously differentiable function over the . Along the real axis outside the interval , the real part of is constant; going away from the real axis into the gives a function that approaches as . 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 in the interval ; going away from the real axis into the is a function that approaches logarithmically. 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 in the interval going away from the real axis into the is a function that approaches logarithmically. 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. Away from the origin, the function absolute value of is logarithmically increasing. 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.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. At the function has a logarithmic 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. At the function has a logarithmic singularity. 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 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 a logarithmic 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 a logarithmic 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 a logarithmic 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 a logarithmic singularity. At a zero from square root branch point exists.
