Argconjugatez21

Argconjugatez21 is a mathematical operator defined on the complex plane. For any nonzero complex number z, Argconjugatez21(z) denotes the principal value of the argument of (conj(z))^21, where Arg denotes the principal value of the argument.

If z = r e^{iθ}, then conj(z) = r e^{-iθ}, and (conj(z))^21 = r^{21} e^{-i21θ}. Therefore Argconjugatez21(z) equals the principal value of -21θ. The result depends only on the argument θ and is independent of the modulus r. As a consequence, Argconjugatez21 maps each ray from the origin to a single angle, with the angle wrapping into the principal range.

Example: take z = 2 e^{iπ/6}. Here θ = π/6, so Argconjugatez21(z) = principal(-21 · π/6) = principal(-3.5π) = π/2.

In theoretical contexts such as Arg-conjugate dynamics, Argconjugatez21 serves as a toy operator to study angular behavior under conjugation and high-power mappings. It relates to basic concepts such as the complex conjugate, the argument function, and power maps, illustrating how angle information is transformed under conjugation and exponentiation.

See also: Complex conjugate, Argument of a complex number, Power map, Angular dynamics.