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 W 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 \(W_k\) of the Lambert W function as a line integral.

中文翻译：

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缠绕数、展开数和朗伯 W 函数

复数的展开数被引入处理涉及复数和多值复函数的自动计算，并已成功应用于涉及兰伯特W函数分支的计算。在这部分说明性说明中，我们从纯粹的拓扑角度讨论展开数，并将其与复平面中曲线的经典缠绕数联系起来。我们还使用展开数将 Lambert W函数的分支\(W_k\)表示为线积分。