Argconjugatez2ii

Argconjugatez2ii is a hypothetical mathematical construct in complex analysis describing a real-valued transformation of a complex number obtained by taking the conjugate of its square and then applying the principal value of the argument function. In short, for a nonzero complex number z, Argconjugatez2ii(z) is defined as Arg(conj(z^2)).

Formal definition: Let z be a nonzero complex number, z = re^{iθ} with r > 0 and θ in the principal range (-π, π]. Then z^2 = r^2 e^{i2θ}, conj(z^2) = r^2 e^{-i2θ}, and Arg(conj(z^2)) is the principal value of the angle of conj(z^2). Equivalently, Argconjugatez2ii(z) = the principal value of -2θ, i.e., the unique number ψ in (-π, π] such that ψ ≡ -2θ (mod 2π).

Properties: The value depends only on the argument θ of z, not on its magnitude r. It is invariant under positive scaling z -> tz with t > 0, since r is canceled in the angle. The function is not injective: many different z can share the same Argconjugatez2ii value. It is continuous on C \ {0} and exhibits a wrap-around behavior wherever -2θ falls outside the principal range.

Examples: If z = 1 (θ = 0), Argconjugatez2ii(z) = 0. If z = i (θ = π/2), Argconjugatez2ii(z) = π. If z = e^{iπ/3} (θ = π/3), Argconjugatez2ii(z) = -2π/3.

Applications of Argconjugatez2ii are primarily theoretical, illustrating how conjugation and squaring interact with the argument function. The term is not a standard operator in mainstream texts and is described here as a representative example of angular transforms in the complex plane.