Argument

The argument of a function is the variable (also called the independent variable) on which the value of the function depends.

The argument of a complex number $z=x+iy=r(\cos\phi+i\sin\phi)$, represented in the plane by the point with coordinates $x$ and $y$, is the angle $\phi$ of the radius vector $r$ of this point with the $x$-axis. A branch of the argument is a specification of the range of values for $\phi$, for example, $0 \le \phi < 2\pi$ or $-\pi < \phi \le \pi$.