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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
We also prove that (D, dT ) and (D, dP ) , where dP denotes the Poincaré metric, have the sameisometry group and then classify the isometries of (D, dT ) .
📌 3 Key Takeaways
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Objective & Hypothesis:The complex unit disk D = {z ∈ C: |z| < 1} is endowed with Möbius addition ⊕M defined by $woplus M;;z=frac{w+z}{1+wz}$ We prove that the metric dT defined on D by $d_t(omega,z)=tan^{-1}left|-omegaoplus M;zright|$ is an invariant of Möbius transformationscarrying D onto itself.
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Key Findings & Data:The complex unit disk D = {z ∈ C: |z| < 1} is endowed with Möbius addition ⊕M defined by $woplus M;;z=frac{w+z}{1+wz}$ We prove that the metric dT defined on D by $d_t(omega,z)=tan^{-1}left|-omegaoplus M;zright|$ is an invariant of Möbius transformationscarrying D onto itself.
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Conclusion & Impact:We also prove that (D, dT ) and (D, dP ) , where dP denotes the Poincaré metric, have the sameisometry group and then classify the isometries of (D, dT ) .
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
The complex unit disk D = {z ∈ C: |z| < 1} is endowed with Möbius addition ⊕M defined by $woplus M;;z=frac{w+z}{1+wz}$ We prove that the metric dT defined on D by $d_t(omega,z)=tan^{-1}left|-omegaoplus M;zright|$ is an invariant of Möbius transformationscarrying D onto itself. We also prove that (D, dT ) and (D, dP ) , where dP denotes the Poincaré metric, have the sameisometry group and then classify the isometries of (D, dT ) .
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Cite This Article
SUKSUMRAN, T., DEMİREL, O. (2022). A metric invariant of Möbius transformations. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 6, 2876–2887. https://ajindex.com/articles/608021