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:
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 word covering:
“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 (18191892)