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].

Read more about this topic:  Argument (complex Analysis)

Famous quotes containing the words covering and/or space:

    “Three forms I see on stretchers lying, brought out there untended
    lying,
    Over each the blanket spread, ample brownish woolen blanket,
    Gray and heavy blanket, folding, covering all.”
    —Walt Whitman (1819–1892)

    “No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
    —Isaac Newton (1642–1727)