Entering the complex plane Upper picture: in the upper half of the complex ‐plane near the real axis viewed from the lower half‐plane.
Lower picture: in the lower half of the complex ‐plane 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. At , the function has a square root branch point. Along the real axis, the real part of approaches as . Along the real axis, the imaginary part of vanishes identically for ; going away from the real axis into the complex ‐plane gives a function that approaches . The imaginary part is discontinuous along the interval . The imaginary part has upper 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 interval the function has a branch cut. The imaginary part has discontinuities along the branch cut. At , the function has a square root branch point. 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 . The viewpoint is from the upper half‐plane. At , the function has a logarithmic singularity. 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 point at is a square root branch point. Only the imaginary part shows discontinuities due to the branch cut. 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 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. The function has a square root branch point at ; going away from the real axis into the upper half of the gives a function that asymptotically approaches ∞. 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 a square root branch point at ; going away from the real axis into the upper half of the gives a function that asymptotically approaches ∞. 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 for ; going away from the real axis into the gives a function that approaches . is a discontinuous function over the . The branch point at is a square root 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. Along the real axis, the imaginary part of vanishes identically for ; going away from the real axis into the gives a function that approaches . is a discontinuous function over the . The branch point at is a square root branch point. 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. Zeropole plot The product 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. 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. 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 product 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.
