SemidiskEdit
In mathematics, a semidisk is the portion of the unit disk that lies above the real axis. Formally, it is the open set D+ = { z ∈ C : |z| < 1 and Im(z) > 0 }. Its boundary consists of two pieces: the circular arc { z ∈ C : |z| = 1, Im(z) > 0 } and the straight line segment { z ∈ R : -1 ≤ z ≤ 1 }. This combination gives a simple, yet nontrivial, domain that is widely used in the theory of complex analysis and related areas. For those who want a familiar reference point, the semidisk sits inside the broader context of the unit disk and its boundary structure, and it is often discussed together with the upper half-plane and the full disk Unit disk.
The semidisk serves as a classic testbed for ideas in conformal mapping, boundary value problems, and spaces of analytic functions. It is a convenient example where one can see how boundary geometry—in this case a straight edge meeting a curved edge at a corner—affects analytic behavior, function spaces, and mapping properties. The topic is closely tied to foundational concepts in Complex analysis, including Conformal mapping, the Riemann mapping theorem, and the Schwarz reflection principle Schwarz reflection principle.
Definition and geometry
- Formal definition: as above, D+ = { z ∈ C : |z| < 1, Im(z) > 0 }.
- Boundary description: ∂D+ = { z ∈ C : |z| = 1, Im(z) ≥ 0 } ∪ { z ∈ R : -1 ≤ z ≤ 1 }. The arc part is a portion of the unit circle, while the diameter part lies on the real axis.
- Topology and symmetry: D+ is a simply connected, open subset of the plane. It is symmetric with respect to reflection across the imaginary axis, since replacing z by −z̄ preserves the defining conditions.
Because the boundary includes a straight line segment meeting a curved arc, D+ is a “piecewise smooth” Jordan domain with a corner at the two endpoints z = −1 and z = 1. The corner angles influence how boundary values of harmonic and analytic functions behave and how conformal maps extend to the boundary.
Analytic structure and mappings
- Conformal maps: D+ is a prime example of a simply connected domain on which one studies conformal equivalence. A simple, explicit map z ↦ z^2 is conformal and injective on D+. This map sends the semidisk onto the unit disk minus the positive real axis, illustrating how boundary geometry can produce nontrivial images while preserving local angle structure. More generally, conformal maps of D+ to other standard domains are analyzed via standard tools such as Möbius transformations and Schwarz–Christoffel theory when mapping to polygons or to half-planes.
- Relation to the unit disk and half-planes: The semidisk sits between the full unit disk and the upper half-plane in the web of classic domains. Through standard transformations, one can transfer questions about analytic functions on D+ to questions on the unit disk Conformal mapping and on the Upper half-plane.
- Function spaces: Because D+ is a model of a nice, simple planar region, it is used to define and study spaces of analytic functions with square-integrable or boundary-valued behavior, such as Hardy space on D+ and related Bergman spaces Bergman space. Techniques from these spaces—Poisson integrals, boundary values, and reproducing kernels—have direct analogues in the semidisk setting Poisson kernel.
- Boundary value phenomena: The presence of the straight boundary segment means that boundary data on −1 ≤ x ≤ 1 interacts with data on the arc. The Schwarz reflection principle, which is most naturally framed for purely real or purely imaginary boundaries, can be adapted in the semidisk context to extend certain harmonic or analytic functions across the straight boundary under symmetry considerations Schwarz reflection principle.
Applications and perspective
- Boundary value problems: The semidisk provides a concrete arena for testing how boundary geometry affects solvability and regularity of boundary value problems for Laplace’s equation and for analytic functions. The interplay between the arc and the diameter makes it a good pedagogical example for understanding how corners influence potential-theoretic measures and harmonic measure.
- Conformal mapping as a constructive tool: By studying the semidisk, one gains concrete intuition for constructing conformal maps from simple domains to more complicated ones. The idea that a relatively simple map can turn a geometrically simple region into a more intricate one—while preserving angles—serves as a cornerstone of several complex variables and geometric function theory Conformal mapping.
- Connections to classical theories: The semidisk appears in discussions of boundary regularity, approximation by rational or polynomial functions, and the behavior of kernels in reproducing kernel Hilbert spaces. The ideas developed in the semidisk context feed back into broader topics such as Harmonic function theory and boundary behavior in Complex analysis.