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Möbius transformation (Definition)

A Möbius transformation is a bijection on the extended complex plane $\mathbb{C} \cup \{\infty\}$ given by

$$f(z) = \begin{cases} \frac{a}{c} &\text{if } z = \infty \\ \infty &\text{if } z = -{d \over c}\\ \frac{az+b}{cz+d} &\text{otherwise} \end{cases} $$

where $a,b,c,d \in \mathbb{C}$ and $ad - bc \ne 0$ It can be shown that the inverse, and composition of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition.

The geometric interpretation of the Möbius group is that it is the group of automorphisms of the Riemann sphere.

Any Möbius map can be composed from the elementary transformations - dilations, translations and inversions. If we define a line to be a circle passing through $\infty$ then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.




"Möbius transformation" is owned by Koro. [ full author list (2) | owner history (1) ]
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See Also: proof of conformal Möbius circle map theorem, automorphisms of unit disk, unit disk upper half plane conformal equivalence theorem, inversion of plane

Other names:  fractional linear transformation, linear fractional transformation

Attachments:
isomorphism of the group PSL_2(C) with the group of Möbius transformations (Result) by rspuzio
automorphisms of unit disk (Example) by brianbirgen
modular group (Definition) by rm50
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Cross-references: transformation, passing through, circle, line, inversions, translations, dilations, transformations, map, Riemann sphere, automorphisms, interpretation, group, composition, inverse, extended complex plane, bijection
There are 14 references to this entry.

This is version 18 of Möbius transformation, born on 2002-02-19, modified 2007-05-22.
Object id is 2181, canonical name is MobiusTransformation.
Accessed 16445 times total.

Classification:
AMS MSC30D99 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Miscellaneous)

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