Winding Numbers, Unwinding Numbers, and the Lambert W Function

Abstract

<jats:title>Abstract</jats:title><jats:p>The unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert <jats:italic>W</jats:italic> function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches <jats:inline-formula><jats:alternatives><jats:tex-math>$$W_k$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> of the Lambert <jats:italic>W</jats:italic> function as a line integral.</jats:p>