argSxy

argSxy is a notation used to denote the argument, or phase angle, of a complex-valued quantity S evaluated at a pair of real variables x and y. When S is a function of two real variables and takes complex values, it can be written as S(x,y) = u(x,y) + i v(x,y), where u and v are real-valued. The argument represents the angle of the complex number S(x,y) in the complex plane and is commonly defined as argSxy = atan2(v(x,y), u(x,y)). The principal value of the argument typically lies in the interval (-π, π], but in general the argument is multivalued, taking the form θ + 2kπ for any integer k.

In practice, argSxy is used to describe the phase of a two-variable complex surface. It is undefined at points where S(x,y) = 0, and in those regions the argument can be discontinuous, leading to branch-like behavior or phase unwrapping considerations in numerical work. If S is analytic in a region, the behavior of its argument interacts with the local properties of S, though the argument itself is not a holomorphic function.

Applications of argSxy appear in fields such as wave physics, optics, and signal processing, where the phase of a complex-valued field or transform provides important information. For example, in image and surface analysis, the map (x,y) -> argSxy conveys the local phase pattern of a complex-valued surface.

Notes: the exact notation argSxy is not universally standardized; some texts use Arg S(x,y) or refer to the phase of S more generally. It is commonly computed via the two-argument arctangent function, atan2.