Conjz2

Conjz2 is a notation that appears in some mathematical texts to denote the complex conjugate of the square of a complex number z. Formally, conjz2(z) = conj(z^2) = (conj z)^2, where conj denotes complex conjugation.

If z = x + iy, then conjz2 = (x − iy)^2 = (x^2 − y^2) − 2ixy. This shows that conjz2 is the complex conjugate of z^2, meaning conj(conjz2) = z^2 and |conjz2| = |z|^2.

Key properties follow from the nature of complex conjugation. Conjugation is an involution and an automorphism of the complex field, so conj(a + b) = conj(a) + conj(b) and conj(ab) = conj(a) conj(b). Consequently, conj(z^2) = (conj z)^2, illustrating that the order of squaring and conjugation can be interchanged in this case.

Geometrically, taking the conjugate corresponds to reflecting a complex number across the real axis. Therefore conjz2 represents the reflection of z^2 in the complex plane. This operation leaves real z unchanged, and for purely imaginary z the square z^2 is real, so conjz2 = z^2 in those instances.

Conjz2 is not universally standardized in notation; many authors simply write conj(z^2) or (conj z)^2. Its practical use lies in analyses where the relationship between squaring and conjugation is relevant, such as simplifying expressions or examining symmetry in complex-valued functions. See also complex conjugate and complex squares.