Argconjugatez1z

Argconjugatez1z is a term used in some mathematical contexts to denote the argument (angle) of the complex conjugate of the product of two complex numbers, z1 and z. It is not a standard or universally adopted operator in mainstream complex analysis, but it appears in certain notes, software documentation, or niche expositions as a compact way to express a specific phase computation.

Definition and interpretation

Let z1 and z be nonzero complex numbers. Write z1 = r1 e^{iθ1} and z = r e^{iθ2}, with θ1 and θ2 representing their arguments (principal values chosen within a fixed range, such as (-π, π]). Then z1 z = (r1 r) e^{i(θ1+θ2)}, and its complex conjugate is conj(z1 z) = (r1 r) e^{-i(θ1+θ2)}. The argument of conj(z1 z) is Arg(conj(z1 z)) = - (θ1 + θ2). Equivalently, Argconjugatez1z can be described as - Arg(z1 z) when the principal value of the argument is used and the result is wrapped to the chosen range.

Properties

- Argconjugatez1z depends only on the arguments of z1 and z, not on their magnitudes.

- It equals the negative of the sum of the individual arguments, up to the chosen branch/wrapping convention.

- If either z1 or z is zero, Argconjugatez1z is undefined due to the undefined argument of zero.

Examples

If z1 = e^{iα} and z = e^{iβ}, then Argconjugatez1z = -(α + β). If z1 and z have arguments 2 and -π/3 respectively, the value is -(2 - π/3) = -2 + π/3, after applying the principal value wrap.

See also

Arg, complex conjugate, polar form, phasor arithmetic.