Argconjugatez2iiz

Argconjugatez2iiz is a speculative mathematical operator described in informal discussions and some teaching materials as a composite operation that blends complex conjugation, the complex argument, and a scaling by 2i. It is not part of standard complex analysis, but is used in examples to illustrate how conjugation and angle information can interact in the complex plane.

Definition and notation: A common illustrative definition treats Argconjugatez2iiz as a map from nonzero complex numbers z to a complex number w given by w = 2i · Arg(conj(z)) · z, where Arg denotes the principal value of the argument and conj(z) is the complex conjugate. It is important to note that this definition is not canonical; different authors may propose variants.

Properties: The operator is nonlinear and not analytic, since it involves the modulus- or angle-based Arg function. It depends on the chosen branch of Arg and is not invariant under simple transformations. In general, Argconjugatez2iiz does not preserve magnitude or direction in a straightforward way, and its behavior intermingles the magnitude of z with the argument of its conjugate.

Example: For z = 1 + i, conj(z) = 1 − i and Arg(conj(z)) = −π/4. Then w = 2i(−π/4)(1 + i) = π/2 − iπ/2.

Notes and see also: Because Argconjugatez2iiz is not a standard operator, definitions vary and are used mainly as teaching aids. It is often discussed alongside the complex conjugate, the argument function, and polar form to explore how conjugation affects angle information and how composite operations can behave in the complex plane.