The map sends the open second quadrant inside the unit circle bijectively to the open first quadrant outside it. Radius becomes and argument becomes . Its derivative is nonzero on the source, so it is a conformal map. Rotation combined with reciprocal inversion similarly maps other quarter-disc regions.
The map is a conformal map from onto the unit disc. A principal square root first maps the slit parabolic region to a vertical strip, and a scaled tangent followed by squaring maps the strip to a slit disc. Evenness in the square-root variable gives a holomorphic extension across the removed axis, filling the target slit.
A prime end describes an approach to the boundary of a simply connected domain through a nested sequence of crosscuts. A conformal map induces a correspondence between prime ends, even when the Euclidean boundary does not have a continuous one-to-one parametrization. This is the natural boundary interpretation of the endpoints and growing tip of a Loewner chain.
Choose a conformal map from a proper simply connected domain to the unit disc and pull back the topology of the closed disc. The resulting compactification is independent of the chosen map, since disc automorphisms extend to its boundary. Its intrinsic boundary distinguishes different approaches to a slit. Every conformal map between such domains extends to their intrinsic compactifications. This boundary agrees with the prime-end description in the simply connected setting.
For a simply connected domain and , the conformal radius is , where maps the unit disc conformally onto with . Disc automorphisms fixing zero are rotations, so the value is independent of the chosen map.
Before the Loewner swallowing time of , set and . Since the conformal radius of the complex upper half-plane at is , the transformation rule gives . The Koebe quarter theorem and the interior-disc comparison give . Differentiating the Chordal Loewner equation gives . Thus this quantity decreases as the remaining domain shrinks.
Two plane domains are conformally equivalent when there is a bijective holomorphic map between them whose inverse is holomorphic.
A conformal map from into the punctured unit disc would be bounded near its isolated singularity at zero, hence extend to a bounded entire function. The Liouville theorem would make it constant.
Composing a hypothetical conformal equivalence with a Cayley transform would give a nonconstant bounded entire function, contradicting the Liouville theorem.
The radial map is a homeomorphism from to the unit disc, with inverse . A conformal equivalence in that direction would be a nonconstant bounded entire function, contradicting the Liouville theorem.
The mapmaps conformally onto the upper half-plane. The Möbius transformation maps the half-disc onto the first quadrant, and squaring maps that quadrant onto the upper half-plane.
If a holomorphic function satisfies , then it has a holomorphic local inverse near and is conformal there. Its possible failures of local conformality are therefore its critical points.
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A **conformal map** is a function between two shapes or spaces that preserves angles locally but may change sizes. In more technical terms, a conformal mapping is a function \( f \) that is holomorphic (complex differentiable) and has a non-zero derivative in a domain of the complex plane. ### Key Properties of Conformal Maps: 1. **Angle Preservation**: Conformal maps preserve the angle between curves at their intersections, which means the local geometric structure is maintained.