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nonequilibrium
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So every Möbius transformation of the complex plane is holomorphic and 1-to-1 on the Riemann sphere. Is the converse also true, or are there counter-examples?
https://en.wikipedia.org/wiki/Möbius_transformationThe Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold
A Möbius transformation is a type of complex function that maps points in the complex plane to other points in the complex plane. It can be represented by a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, and d are complex numbers and ad - bc ≠ 0.
A Möbius transformation is a holomorphic function, which means it is complex-differentiable at every point in its domain. This is because the rational function form of a Möbius transformation satisfies the Cauchy-Riemann equations, which are necessary conditions for a function to be holomorphic.
A 1-to-1 Möbius transformation is one that maps each point in its domain to a unique point in its range. This means that the transformation is invertible, and its inverse is also a Möbius transformation. This property is important in understanding the geometric properties of Möbius transformations.
No, not all holomorphic functions can be represented as Möbius transformations. In order for a function to be a Möbius transformation, it must satisfy certain conditions, such as being a rational function and having a non-zero determinant of its coefficient matrix. Many holomorphic functions do not meet these criteria and cannot be represented as Möbius transformations.
Möbius transformations have many applications in mathematics and science, including complex analysis, differential geometry, and fluid dynamics. They are also used in computer graphics and image processing to create distortions and transformations of images. In physics, they have been used to model the behavior of objects moving in circular or elliptical orbits. Additionally, Möbius transformations have connections to other areas of mathematics, such as group theory and topology.