realaxis

Realaxis refers to the real axis in the complex plane. It is the set of all complex numbers with zero imaginary part and can be written as {x ∈ R} or {x + 0i | x ∈ R}. In a standard Argand diagram, it corresponds to the horizontal axis and is often identified with the real number line.

As a subset of the complex plane, the Realaxis is a one-dimensional real manifold and is topologically homeomorphic to the real line. It is closed in the complex plane and forms the natural reference line for many geometric and analytic constructions. Points on the Realaxis have the form z = x, with y = 0, and correspond directly to real numbers.

Operations and transformations: Adding a real number or multiplying by a real scalar maps the Realaxis into itself. Complex conjugation fixes the Realaxis pointwise. The Realaxis also serves as the boundary between the upper and lower halves of the complex plane, a role that is important in various results of complex analysis, such as reflection principles and boundary value problems.

Applications: In visualization, the Realaxis is the x-axis of the Argand plane. It provides a reference for studying limits, real-valued functions embedded in complex analysis, and the behavior of analytic mappings near the boundary.

See also: Complex plane, imaginary axis, conjugation, Arg, reflection principle.