Complex Analysis
Riemann Mapping Theorem
Draft for Information Only ContentRiemann Mapping Theorem source/reference: Riemann Mapping TheoremConformal MappingsProperties
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And what conformal mappings are ther of the form 𝑓:𝔻→𝐷, where 𝔻=𝐵_{1}(0) is the unit disk and 𝐷⊂ℂ? The Riemann Mapping TheoremBy theorem. If 𝐷 is a simply connected domain (= open, connected, no holes) in the complex plane, but not the entire complex plane, then there is a conformal map (= analytic, onetoone, onto) of 𝐷 onto the open unit disk 𝔻. That is "𝐷 is conformally equivalent to 𝔻" The Riemann MapLet 𝐷 be a simply connected domain. In order to find a unique conformal mapping 𝑓 from 𝐷 onto 𝔻, "3 real parmeeters" are needed to specify. For example, specify
The Upper Half PlaneLet 𝐷 be the upper half plane, i.e. 𝐷={𝑧: Then the line through 0, 1, ∞ (the real axis) must be mapped to the circle through 1, 𝑖, −1 (the unit circle). Further, the domain to the left of the real axis (𝐷) is then mapped to the domain to the left of the unit circle (𝔻), oriented by the ordering of the given points. Finding the Riemann Map The restriction of the Mobius transformation 𝑓 to the upper half plane 𝐷 thus maps 𝐷 onto 𝔻. Finding a formula for 𝑓, such that 𝑓 maps 0, 1, ∞ to 1, 𝑖, −1.
Thus 𝑓(𝑧)= The first quadrant to the upper half of the unit disk Let 𝑄 be the first quadrant, i.e. the domain in the complex plane, bounded by the positive real axis and the positive imaginary axis. Since the map 𝑓 maps 0 to 1, 𝑖 to 0, and ∞ to −1, it maps the line through 0, 𝑖, ∞ (i.e. the imaginary axis) to the line through 1, 0, −1 (i.e. the real axis). Hence the restriction of 𝑓 to 𝑄 maps 𝑄 conformally onto the upper half of the unit disk, 𝔻^{+} The first quadrant to the upper half plane The map 𝑔(𝑧)=𝑧^{2} is injective and analytic in the first quadrant 𝑄 𝑔 maps 𝑄 conformally onto its image, namely the upper half plane 𝐷 The Riemann Map of the upper half of the unit disk The previous three examples help to construct the Riemann map from 𝔻^{+} to 𝐷:
ApplicationMany problems are easier to solve in the unit disk (or some other "nice" standard region) than in the region they are formulated in. Solutions can be found in the standard region, then transported back to the original region via a Riemann map Example: Fluid flow can be modeled nicely in the upper half plane. To understand a similar fluid flow in another region, map this flow from the upper half plane to the desired region using the Riemann map. Other examples: electrostatics, heat conduction, aerodynamics, etc. ©sideway ID: 190400029 Last Updated: 2019/4/29 Revision: 
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