complexargument

The complex argument of a nonzero complex number z is the angle θ that z makes with the positive real axis in the complex plane. If z is written in polar form as z = r e^{i θ} or z = r (cos θ + i sin θ), with r = |z| ≥ 0, then θ is the argument of z. The angle is defined modulo 2π, so all arguments of z are of the form θ + 2πk, where k is any integer. The principal value of the argument, denoted Arg z, is the unique θ in a fixed interval (commonly (-π, π] or [0, 2π)) such that z ≠ 0 and z/|z| = e^{i θ}.

To compute the argument from Cartesian coordinates z = x + i y, one can use θ = atan2(y, x), which accounts for the correct quadrant and avoids division-by-zero issues. The modulus r = |z| = sqrt(x^2 + y^2) together with θ gives the polar form z = r e^{i θ} or z = r (cos θ + i sin θ).

Properties and consequences:

- Multiplication adds arguments: Arg(z1 z2) = Arg z1 + Arg z2 (mod 2π).

- Division subtracts arguments: Arg(z1 / z2) = Arg z1 − Arg z2 (mod 2π).

- Powers scale the argument: Arg(z^n) = n Arg z (mod 2π).

- The argument is not defined for z = 0, since the angle is undefined at the origin.

Notes:

- Some authors use arg z to denote the multi-valued set {θ + 2πk}, while Arg z designates the principal value.

- The principal value introduces a branch cut along the chosen interval’s boundary, commonly the negative real axis.