Argconjugatez2z

Argconjugatez2z is a hypothetical mathematical operator defined for nonzero complex numbers z. It maps z to the principal value of the argument of the complex conjugate raised to the third power, that is, Argconjugatez2z(z) = Arg(conj(z)^3), where Arg denotes the principal value of the argument.

If z is written in polar form as z = r e^{iθ} with r > 0 and θ in the standard principal range, then conj(z) = r e^{-iθ}, and conj(z)^3 = r^3 e^{-i3θ}. Therefore Argconjugatez2z(z) = Arg(r^3 e^{-i3θ}) equals the principal value of -3θ. Equivalently, Argconjugatez2z(z) ≡ -3 Arg(z) (mod 2π), with the result brought back to the principal value interval.

Domain and range: The domain is C \ {0}, and the range is typically the interval (-π, π], depending on the chosen convention for Arg.

Key properties: The value of Argconjugatez2z(z) depends only on the argument of z, not its magnitude. It is invariant under scaling z by a positive real factor. It differs from Arg(z^3), which equals 3 Arg(z); Argconjugatez2z uses the complex conjugate and introduces the negative triple of the argument, followed by a wrapping to the principal value.

Examples:

- z = 1 (θ = 0) gives Argconjugatez2z(1) = 0.

- z = i (θ = π/2) gives Argconjugatez2z(i) = -3π/2, wrapped to π/2.

- z = -1 (θ = π) gives Argconjugatez2z(-1) = π.

See also: Arg, complex conjugate, principal value of the argument, z.