Understanding Branch Cuts of Expressions

TL;DR

This paper presents methods and a Maple package for calculating and visualizing the complex branch cuts of expressions involving multi-valued functions, aiding in their analysis and computation.

Contribution

It introduces explicit techniques and a software package for computing and visualizing branch cuts of complex expressions in computer algebra systems.

Findings

Developed algorithms for branch cut calculation

Implemented the techniques in Maple's BranchCuts package

Enhanced understanding of complex expression cuts

Abstract

We assume some standard choices for the branch cuts of a group of functions and consider the problem of then calculating the branch cuts of expressions involving those functions. Typical examples include the addition formulae for inverse trigonometric functions. Understanding these cuts is essential for working with the single-valued counterparts, the common approach to encoding multi-valued functions in computer algebra systems. While the defining choices are usually simple (typically portions of either the real or imaginary axes) the cuts induced by the expression may be surprisingly complicated. We have made explicit and implemented techniques for calculating the cuts in the computer algebra programme Maple. We discuss the issues raised, classifying the different cuts produced. The techniques have been gathered in the BranchCuts package, along with tools for visualising the cuts. The package is included in Maple 17 as part of the FunctionAdvisor tool.