righthalfplane

The right half-plane, denoted here as the right half-plane and sometimes abbreviated as RHP, is the set of complex numbers z = x + i y with real part x > 0. Its boundary is the imaginary axis, consisting of all points with real part equal to zero. The complementary half-plane consists of points with x < 0.

Geometrically, the right half-plane is an open, unbounded region in the complex plane. It is connected and simply connected, making it a standard domain for complex analysis. It is invariant under translations by purely imaginary numbers and under scaling by positive real numbers.

A fundamental property is its conformal equivalence to the open unit disk. There exists a Möbius transformation, for example w = (z − 1)/(z + 1), that maps the right half-plane onto the unit disk, sending the imaginary axis to the unit circle. Conversely, the disk can be mapped back to the right half-plane, illustrating the deep connection between these canonical domains.

Automorphisms of the right half-plane can be described in terms of Möbius transformations with real coefficients, reflecting its boundary along the imaginary axis. The right half-plane often serves as a natural domain for problems in complex analysis, including Hardy spaces and potential theory.

In applied contexts, the right half-plane appears as the region of convergence for certain Laplace transforms and as the s-plane domain used in control theory and signal processing, where poles with positive real part indicate instability.

See also: left half-plane, unit disk, Möbius transformation, conformal mapping.