Argconjugate

Argconjugate is a mathematical transformation applied to complex numbers. It is defined as the operation that preserves the modulus and negates the argument; for a complex number z = r e^{iθ} with r ≥ 0 and θ ∈ R, Argconjugate(z) = r e^{-iθ} = z̄, the complex conjugate of z. In standard treatments, Argconjugate coincides with the complex conjugation map.

Properties and implications. Argconjugate is an involution: applying it twice returns the original number, Argconjugate(Argconjugate(z)) = z. It preserves modulus, since |Argconjugate(z)| = |z|, and it reflects the number across the real axis, changing the sign of the argument. For real numbers, Argconjugate(z) = z. It interacts with multiplication by the identity conj(z1 z2) = conj(z1) conj(z2); hence Argconjugate distributes over multiplication in the sense that Argconjugate(z1 z2) = Argconjugate(z1) Argconjugate(z2). For real scalars a, Argconjugate(a z) = a Argconjugate(z); for complex scalars, Argconjugate(a z) = ā Argconjugate(z). The map is anti-holomorphic and conjugate-linear, reflecting the standard analytic properties of complex conjugation.

In polar coordinates, the operation corresponds to (r, θ) → (r, −θ). This makes Argconjugate a natural tool for studying symmetry, reflections, and mirror-image phenomena in the complex plane. It is frequently used in discussions of complex-valued signals, root-finding with conjugate pairs, and geometric interpretations of complex functions.

See also: complex conjugate, polar form, reflection across the real axis. Examples: Argconjugate(3 + 4i) = 3 − 4i; Argconjugate(e^{iθ}) = e^{−iθ}.