Lecture Notes of Möbuis Transformation in Hyperbolic Plane

Affiliation(s)

Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

ABSTRACT

In this paper, I have provided a brief introduction on M?bius transformation and explored some basic properties of this kind of transformation. For instance, M?bius transformation is classified according to the invariant points. Moreover, we can see that M?bius transformation is hyperbolic isometries that form a group action PSL (2, R) on the upper half plane model.

KEYWORDS

The Upper Half-Plane Model, Möbius Transformation, Hyperbolic Distance, Fixed Points, The Group PSL (2, R)

The Upper Half-Plane Model, Möbius Transformation, Hyperbolic Distance, Fixed Points, The Group PSL (2, R)

Cite this paper

Amer, R. (2014) Lecture Notes of Möbuis Transformation in Hyperbolic Plane.*Applied Mathematics*, **5**, 2216-2225. doi: 10.4236/am.2014.515215.

Amer, R. (2014) Lecture Notes of Möbuis Transformation in Hyperbolic Plane.

References

[1] Yilmaz, N. (2009) On Some Mapping Properties of Möbius Transformations. The Australian Journal of Mathematical Analysis and Applications, 6, 1-8.

[2] Nehari, Z. (1952) Conformal Mapping. McGraw-Hill Book, New York.

[3] John, O. (2010) The Geometry of Möbius Transformations. University of Rochester, Rochester.

[4] Graeme, K.O. (2012) Random Discrete Groups in the Space of Möbuis Transformations. Msc Thesis, Massey University, Albany.

[5] Aramayona, J. (2011) Hyperbolic Structures on Surfaces. Lecture Notes Series, IMS, NUS, 9.

[6] Beardon, A.F. (1995) The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91. Springer-Verlag, New York.

[7] Jones, G.A. and Singerman, D. (1987) Complex Functions, an Algebraic and Geometric Viewpoint. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9781139171915

[8] Lehner, J. (1964) Discontinuous Groups and Automorphic Functions, Mathematical Surveys, 8. American Mathematical Society, Providence. http://dx.doi.org/10.1090/surv/008

[9] Seppälä, M. and Sorvali, T. (1992) Geometry of Riemann Surfaces and Teichmüller Spaces, Chapter 1. Elsevier Science Publishing Company, INC, New York, 11-58.

[1] Yilmaz, N. (2009) On Some Mapping Properties of Möbius Transformations. The Australian Journal of Mathematical Analysis and Applications, 6, 1-8.

[2] Nehari, Z. (1952) Conformal Mapping. McGraw-Hill Book, New York.

[3] John, O. (2010) The Geometry of Möbius Transformations. University of Rochester, Rochester.

[4] Graeme, K.O. (2012) Random Discrete Groups in the Space of Möbuis Transformations. Msc Thesis, Massey University, Albany.

[5] Aramayona, J. (2011) Hyperbolic Structures on Surfaces. Lecture Notes Series, IMS, NUS, 9.

[6] Beardon, A.F. (1995) The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91. Springer-Verlag, New York.

[7] Jones, G.A. and Singerman, D. (1987) Complex Functions, an Algebraic and Geometric Viewpoint. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9781139171915

[8] Lehner, J. (1964) Discontinuous Groups and Automorphic Functions, Mathematical Surveys, 8. American Mathematical Society, Providence. http://dx.doi.org/10.1090/surv/008

[9] Seppälä, M. and Sorvali, T. (1992) Geometry of Riemann Surfaces and Teichmüller Spaces, Chapter 1. Elsevier Science Publishing Company, INC, New York, 11-58.