Argument of Complex Number

Definition

Given a complex number z{0}, the principal argument of z is defined an the real number π<θπ such that

z=r(cos(θ)+isin(θ)).

We write Arg(z)=θ.

Geometrically, the argument represents the angle, measured from the positive real axis, that z forms on an Argand diagram.

More generally, we write arg(z) to either represent the set of choices for θ as above, or for a particular arbitrarily chosen value. However, the somewhat less common notation of argθ for the argument satisfying θ<argθ(z)θ+2π is very useful.

Theorem
argθ(z)=Arg(z)+2nπ

for some nZ.

This result, along with the fact that the principal argument is uniquely defined, follow from the equivalent fact from polar coordinates.