argZ

argZ is an operator or notation used to denote the argument, or angular coordinate, of a complex number Z. The argument is the angle between the positive real axis and the line from the origin to Z. A nonzero complex number Z can be written as Z = r e^{iθ} with r > 0 and θ being an argument of Z.

There is a distinction between the multivalued argument and its principal value. Some texts use Arg(Z) to denote the principal value of the argument, typically in the interval (-π, π], while argZ or Arg may denote the full set of arguments θ + 2πk for integers k. Usage varies by author, and some authors reserve Arg for the principal value and argZ for the multivalued set.

Calculation and relation to polar form: If Z = x + iy with (x, y) ≠ (0, 0), the principal argument is θ = atan2(y, x), and Z = r e^{iθ} with r = sqrt(x^2 + y^2). The usual multivalued Arg(Z) consists of θ + 2πk for k ∈ Z. In this view, θ is one representative of the set of arguments associated with Z.

Properties and notes: argZ is undefined at Z = 0. For nonzero Z1 and Z2, Arg(Z1 Z2) = Arg(Z1) + Arg(Z2) (mod 2π). Arg(conjugate Z) = -Arg(Z). In practice, argZ appears in complex analysis, signal processing, and computer graphics when describing orientation, rotations, or phase. Examples: argZ(1 + i) = π/4; argZ(-1) = π (principal value).