Argconjugatez1

Argconjugatez1 is a geometric transformation on the complex plane defined with respect to a fixed reference point z1. For any complex number z, the operator Argconjugatez1 returns a new point T(z) given by T(z) = z1 + conjugate(z - z1). In practical terms, this operation reflects the vector from z1 to z across the real axis, and then translates it back by z1.

In coordinates, if z1 = x1 + i y1 and z = x + i y, then Argconjugatez1(z) = x + i(2 y1 - y). Thus, Argconjugatez1 fixes every point on the horizontal line through z1 (the line where the imaginary part equals y1) and sends each other point to its mirror image across that line.

Key properties include:

- Involution: applying the transformation twice yields the original point, i.e., Argconjugatez1(Argconjugatez1(z)) = z.

- Isometry: it preserves distances, since it is a reflection in the plane.

- Antiholomorphic: the map is not holomorphic; it involves complex conjugation.

- Geometric interpretation: it is the reflection of the plane across the horizontal line through z1.

Example: Let z1 = 2 + 3i and z = 4 + i. Then Argconjugatez1(z) = 2 + 3i + conjugate((4 - 2) + i(1 - 3)) = 2 + 3i + (2 - i(-2)) = 4 + 5i? More straightforwardly, using the coordinate form, Argconjugatez1(4 + i) = 4 + i(2·3 − 1) = 4 + i5, which lies symmetrically across the line y = 3.

See also: complex conjugation, reflections in the complex plane, involutions, isometries.