The Life of Ryan

Eisenstein Series

Fun with domain colouring of a complex function

Explanation

The Eisenstein series is an example of a modular form, functions defined on the complex numbers with positive imaginary components satisfying certain periodic properties. Specifically, a modular form is a function \(f:\{\tau \in \mathbb{C} \mid \mathrm{Im}(\tau) \gt 0\} \rightarrow \mathbb{C}\) such that:

• \(f\) is holomorphic
• \(f(\tau + 1) = f(\tau)\)
• \(f(-\frac{1}{\tau}) = \tau^wf(\tau)\) for some constant weight \(w\)

The Eisenstein series that is visualized above can be described by the following equation: \[\large{G_{2k}(\tau)=}\Large{\sum_{(m,n) \in \mathbb{Z}^2 \backslash (0,0)}}\large{{\frac{1}{(m+n\tau)^{2k}}}}\] with a weight given by \(k = 4\).

Domain colouring is a practice for visualizing functions with complex-valued outputs. Typically, one maps the argument/phase of the complex-value to the final colour, and its magnitude to the brightness.