Argconjugatez2

Argconjugatez2 is a function defined on the nonzero complex numbers that assigns to each complex number z the principal value of the argument of the square of its complex conjugate. Formally, Argconjugatez2(z) = Arg((conj z)^2), where Arg denotes the principal value of the argument, typically taken in (-π, π].

If z is written in polar form z = r e^{iθ} with r > 0 and θ ∈ ℝ, then Argconjugatez2(z) is the principal value of -2θ. Equivalently, Argconjugatez2(z) = -2 Arg(z) (taken modulo 2π and reduced to the principal interval). The domain is C \ {0}, since the argument is undefined at z = 0, and the range is the interval (-π, π].

Key properties include: for a rotation z → z e^{iα}, Argconjugatez2(z e^{iα}) = Argconjugatez2(z) − 2α (mod 2π); multiplying z by a positive real number does not change the angle, so the magnitude of z does not affect Argconjugatez2. The function reflects the combined effect of complex conjugation (which negates the angle) and squaring (which doubles the angle).

Examples: z = 1 gives Argconjugatez2(1) = 0; z = i yields Argconjugatez2(i) = π; z = −1 yields Argconjugatez2(−1) = 0; z = 1 + i (θ = π/4) yields Argconjugatez2(z) = −π/2.

Relation to related concepts: it is the composition of conjugation and squaring acting on the argument. It is primarily of theoretical interest and may appear in expository or pedagogical discussions rather than as a standard, widely used function.