Entering the complex plane in the near the real axis. 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 function along the real axis. Along the real axis, the real part of is oscillating; going away from the real axis into the its absolute value is generically an exponentially increasing function. Along the real axis, the imaginary part of vanishes identically; going away from the real axis into the its absolute value is generically an exponentially increasing function. 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, is oscillating; going away from the real axis into the its absolute value is generically an exponentially increasing function. Along the lines with , the real part of vanishes identically. 
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 absolute value 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, is oscillating; going away from the real axis into the its absolute value is generically an exponentially increasing function. Along the lines with , the real part of vanishes identically. 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 vanishes identically; going away from the real axis into the its absolute value is generically an exponentially increasing function. Along the lines with , the real part of vanishes identically. 
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 vanishes identically; going away from the real axis into the its absolute value is generically an exponentially increasing function. Along the lines with , the real part of vanishes identically. 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. Along the real axis, is bounded; going away from the real axis gives a exponentially increasing function. 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 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. 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. The essential singularity at results in a complicated structure that cannot be resolved graphically. 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 essential singularity at results in a complicated structure that cannot be resolved graphically. 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 essential singularity at results in a complicated structure that cannot be resolved graphically. 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. The essential singularity at results in a complicated structure that cannot be resolved graphically. 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. The essential singularity at results in a complicated structure that cannot be resolved graphically. 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 essential singularity at results in a complicated structure that cannot be resolved graphically. 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. The essential singularity at results in a complicated structure that cannot be resolved graphically.
