Complex Functions
Interactive visualization of complex functions

Complex
Functions

A modern interactive page for visualizing complex functions: phase is color-coded, while the absolute value appears as contour lines in the domain.

Visualization of the function f(z)=z with colored phase and contour lines for the absolute value
Sample image for the identity function f(z)=z: color represents phase, contour lines represent absolute value.

The idea

Real-valued functions are often visualized by drawing their graphs. This works well on a sheet of paper. For complex functions, however, where both the argument and the value of the function are complex numbers, this is no longer possible; one would need a four-dimensional sheet of paper.

Therefore, complex functions are visualized here in a different way. At each point of a rectangular region of the domain, the corresponding function value is plotted. Its phase is shown in color and its absolute value by contour lines. The contour lines use shades of gray: the larger the absolute value, the brighter the corresponding contour line.

Phase as color The direction of the complex function value is made visible through the color palette.
Absolute value as contour lines The contour lines show how large the absolute value of the function value is.
Computed interactively The visualization is generated directly in an embedded SageMath cell.

Using the interface

Use the sliders to adjust the minimum and maximum values for the real part x and the imaginary part y of the domain. Enter the function you want to visualize in the text box.

Use SageMath notation for mathematical expressions, for example exp(I*z+2*z^2) for exp(iz+2z²). Python notation, such as exp(I*z+2*z**2), is also possible. The variable is always called z.

You can also change the number of contour lines, turn contour labels on or off, and choose between different accuracy levels for the computation. Low accuracy is fastest, but artifacts can sometimes occur, especially in the contour lines. It is recommended to start with low accuracy and increase the accuracy only when needed.

Clicking “Update” computes a new visualization with the current parameter values. Below the output, you will find links to download the current image as a PNG or PDF file.

Interactive visualization

Choose the region, the number of contour lines, and the desired function.

SageMath cell

Notes

This page uses the open-source computer algebra system SageMath. The computations are run on the Sage Cell Server.

If the connection to the Sage Cell Server drops because the server is overloaded, which is visible when the Update field is grayed out, please reload the web page.