Argument (complex Analysis) - Covering Space

Covering Space

In informal situations, arg may be left not well-defined, for instance arg z(t) where z depends on a parameter t may change by 2π every time z goes around the origin. This idea can be made more precise by considering z(t) as being defined not on the complex plane but on a covering space. Polar coordinates excluding the origin and with an unconstrained angle provide such a space, in this case arg is defined by:

\begin{align} \arg \colon \mathbb{R}^+\times\mathbb{R} &\to \mathbb{R} \\ (r,\ \phi) &\mapsto \phi \end{align}

The covering space has as base space the punctured complex plane. This is equivalent to the product of a positive non-zero radius and an angle on a unit circle that is:

The principal value Arg then maps the unit circle component of this representation to the interval (−π, π].

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