Principal Value
Because a complete rotation around 0 leaves a complex number unchanged, there are many choices which could be made for φ by circling the origin any number of times. This is shown in figure 2, a representation of the multi-valued (set-valued) function, where a vertical line cuts the surface at heights representing all the possible choices of angle for that point.
When a well-defined function is required then the usual choice, known as the principal value, is the value in the open-closed interval (−π, π], that is from −π to π radians, excluding −π itself (−180 to +180 degrees). This represents an angle of up to half a complete circle from the positive real axis in either direction, the angle φ is constrained to lie between −π and π radians. This portion of the surface is shown hatched in red in figure 2, and projected onto the plane in figure 4.
The principal value sometimes has the initial letter capitalized as in Arg z, especially when a general version of the argument is also being considered. Note that notation varies, so arg and Arg may be interchanged in different texts.
Some authors define the range of the principal value as being in the closed-open interval [0, 2π).
The set of all possible values of the argument can be written in terms of Arg as:
- .
Read more about this topic: Argument (complex Analysis)
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