Continued fractions: Continued fraction, Pells equation, Mathematical constants, Möbius transformation, Generalized continued fraction, Incomplete gamma function, Gausss continued fraction, Padé table, Stern¿Brocot tree, Silver ratio

Source: Wikipedia. Pages: 50. Chapters: Continued fraction, Pell's equation, Mathematical constants, Möbius transformation, Generalized continued fraction, Incomplete gamma function, Gauss's continued fraction, Padé table, Stern¿Brocot tree, Silver ratio, Minkowski's question mark function, Solving quadratic equations with continued fractions, Convergence problem, Periodic continued fraction, Khinchin's constant, Gauss¿Kuzmin¿Wirsing operator, Padé approximant, Engel expansion, Euler's continued fraction formula, Complete quotient, Restricted partial quotients, Rogers¿Ramanujan continued fraction, Gauss¿Kuzmin distribution, Convergent, Stieltjes transformation, Fundamental recurrence formulas, Chain sequence, Lévy's constant, Lochs' theorem. Excerpt: In geometry, a Möbius transformation of the plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad - bc ¿ 0. Möbius transformations are named in honor of August Ferdinand Möbius, although they are also called homographic transformations, linear fractional transformations, or fractional linear transformations. Möbius transformations are defined on the extended complex plane (i.e. the complex plane augmented by the point at infinity): This extended complex plane can be thought of as a sphere, the Riemann sphere, or as the complex projective line. Every Möbius transformation is a bijective conformal map of the Riemann sphere to itself. Indeed, every such map is by necessity a Möbius transformation. The set of all Möbius transformations forms a group under composition called the Möbius group. It is the automorphism group of the Riemann sphere (when considered as a Riemann surface) and is sometimes denoted .The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds. In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory. Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Klei