Domain Coloring for Common Functions in Complex Analysis

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A domain coloring or phase portrait is a popular and attractive way to visualize functions of a complex variable. The absolute value (modulus) of the function at a point is represented by brightness (dark for small modulus, light for large modulus), while the argument (phase) is represented by hue (red for positive real values, yellow-green for positive imaginary values, etc.). This Demonstration creates domain colorings for many well-known functions in complex analysis: trigonometric and exponential functions, Möbius transformations, the Joukowski map, the Koebe function, Weierstrass functions, gamma and beta functions, and the Riemann zeta function.

Contributed by: Matthew Romney (March 2016)
Open content licensed under CC BY-NC-SA



The terms domain coloring or phase portrait refer to the type of visualization technique for functions of a complex variable used here, or some variation thereof. For a function of a complex variable, each point in the complex plane is colored according to the value of . The brightness represents the absolute value (or modulus) of , with light representing large absolute value and dark representing small absolute value. The hue represents the argument (or phase) of , with red representing positive real numbers, yellow-green representing positive imaginary numbers, cyan representing negative real numbers, and so forth.

In addition to the standard polynomial, trigonometric, and exponential functions, we have included: the Möbius transformation , the Joukowski map , the Koebe function , a few instances of the beta functions , the gamma functions and , and the Weierstrass ℘ functions defined by if , and finally the famous Riemann zeta function , defined as the analytic continuation of the Dirichlet series for .


[1] E. Wegert, Visual Complex Functions: An Introduction with Phase Portraits, Basel: Birkhäuser/Springer Basel AG, 2012.

[2] A. Sandoval-Romero and A. Hernández-Garduño, "Domain Coloring on the Riemann Sphere," The Mathematica Journal, 2015. doi:10.3888/tmj.17-9.

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