# Complex Domain Coloring

Posted on February 15, 2014

## Table of Contents

Real-valued functions $f : \R \rightarrow \R$ with real inputs can be easily visualized on a two-dimensional graph. Real-valued complex functions $f : \C \rightarrow \R$ have two input dimensions and one output dimension, so can be visualized as a three-dimensional surface. Functions $f : \C \rightarrow \C$ with both complex inputs and complex outputs have four dimensions to consider, making them difficult to visualize directly. One popular visualization technique for complex functions is **domain coloring**, which uses color to represent the value a function takes at each point in the complex plane. Domain coloring can help us build visual intuition about complex analysis.