Fractional-linear mapping

fractional-linear transformation

A mapping of the complex space $ \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $ realized by fractional-linear functions (cf. Fractional-linear function).

In the case of the complex plane $ \mathbf C ^ {1} = \mathbf C $, this is a non-constant mapping of the form

$$ \tag{1 } z \rightarrow w = L ( z) = \frac{a z + b }{c z + d } , $$

where $ a d - b c \neq 0 $; the unimodular normalization $ a d - b c = 1 $ is often employed. Any fractional-linear mapping can be additionally defined by the correspondence $ \infty \rightarrow a / c $ and $ - d / c \rightarrow \infty $ to yield a one-to-one mapping of the extended plane $ \overline{\mathbf C}\; $ onto itself. The simplest fractional-linear mappings are the linear ones, $ z \rightarrow w = \widetilde{a} z + \widetilde{b} $, which are obtained if $ c = 0 $. All non-linear fractional-linear mappings can be represented as compositions of two linear mappings and of the mapping $ L _ {0} $: $ z \rightarrow w = 1 / z $. The properties of the fractional-linear mapping $ L _ {0} $ can be illustrated on the Riemann sphere, since if the stereographic projection is employed, it corresponds to the rotation of the sphere through 180° around the diameter passing through the images of the points $ \pm 1 \in \mathbf C $.

Special properties. A fractional-linear mapping maps $ \overline{\mathbf C}\; $ onto itself, conformally and bijectively. The circle property: Under a fractional-linear mapping any circle in $ \overline{\mathbf C}\; $( i.e. a circle in $ \mathbf C $ or a straight line supplemented by the point $ \infty $) transforms into a circle in $ \overline{\mathbf C}\; $. The invariance of the ratio of two symmetrically-located points: A pair of points $ z , z ^ {*} $ which is symmetric with respect to any circle in $ \overline{\mathbf C}\; $ becomes, as a result of a fractional-linear mapping, a pair of points $ w , w ^ {*} $ which is symmetric with respect to the image of this circle. The cross ratio between four points in $ \overline{\mathbf C}\; $ is invariant with respect to a fractional-linear mapping, i.e. if that mapping transforms the points $ \xi _ {1} , \xi _ {2} , \xi _ {3} , \xi _ {4} $ into the points $ \zeta _ {1} , \zeta _ {2} , \zeta _ {3} , \zeta _ {4} $, respectively, then

$$ \tag{2 } \frac{\xi _ {3} - \xi _ {1} }{\xi _ {3} - \xi _ {2} } : \frac{\xi _ {4} - \xi _ {1} }{\xi _ {4} - \xi _ {2} } = \frac{\zeta _ {3} - \zeta _ {1} }{\zeta _ {3} - \zeta _ {2} } : \frac{\zeta _ {4} - \zeta _ {1} }{\zeta _ {4} - \zeta _ {2} } . $$

For any given triplets $ \xi _ {1} , \xi _ {2} , \xi _ {3} $ and $ \zeta _ {1} , \zeta _ {2} , \zeta _ {3} $ of pairwise distinct points in $ \overline{\mathbf C}\; $ there exists one and only one fractional-linear mapping which transforms $ \xi _ {k} \rightarrow \zeta _ {k} $, $ k = 1 , 2 , 3 $, respectively. This fractional-linear mapping can be found from equation (2) by substituting in it $ z $ and $ w $ for $ \xi _ {4} $ and $ \zeta _ {4} $, respectively. The group property: The set of all fractional-linear mappings forms a non-commutative group with respect to composition $ ( L _ {1} L _ {2} ) ( z) = L _ {1} ( L _ {2} ( z) ) $, with unit element $ E ( z) = z $. The universality property: Any conformal automorphism of $ \overline{\mathbf C}\; $ is a fractional-linear mapping and therefore the group of all fractional-linear mappings coincides with the group $ \mathop{\rm Aut} \overline{\mathbf C}\; $ of all conformal automorphisms of $ \overline{\mathbf C}\; $.

All conformal automorphisms of the unit disc $ B = \{ {z \in \mathbf C } : {| z | < 1 } \} $ form a subgroup $ \mathop{\rm Aut} B $ of the group $ \mathop{\rm Aut} \overline{\mathbf C}\; $, consisting of fractional-linear mappings of the type

$$ z \rightarrow w = e ^ {i \theta } \frac{z - \alpha }{1 - \overline \alpha \; z } ,\ | \alpha | < 1 ,\ \mathop{\rm Im} \theta = 0 . $$

The same applies to the conformal automorphisms of the upper half-plane $ \{ {z \in \mathbf C } : { \mathop{\rm Im} z > 0 } \} $; they have the form

$$ z \rightarrow w = \frac{a z + b }{c z + d } ,\ \mathop{\rm Im} ( a , b ,\ c , d ) = 0 ,\ a d - b c > 0 . $$

All conformal homeomorphisms of the upper half-plane onto the unit disc have the form

$$ z \rightarrow w = e ^ {i \theta } \frac{z - \beta }{z - \overline \beta \; } ,\ \ \mathop{\rm Im} \beta > 0 ,\ \mathop{\rm Im} \theta = 0 . $$

Except for the identity fractional-linear mapping $ E ( z) $, fractional-linear mappings have at most two distinct fixed points $ \xi _ {1} $, $ \xi _ {2} $ in $ \overline{\mathbf C}\; $. If there are two fixed points $ \xi _ {1} \neq \xi _ {2} $, the family $ \Sigma $ of circles passing through $ \xi _ {1} $ and $ \xi _ {2} $ is transformed by the fractional-linear transformation (1) into itself. The family $ \Sigma ^ \prime $ of all circles orthogonal to the circles of $ \Sigma $ is also transformed into itself. Three cases are possible in this connection.

1) Each circle of $ \Sigma $ is transformed into itself; such a fractional-linear mapping is said to be hyperbolic and is representable in normal form

$$ \tag{3 } \frac{w - \xi _ {1} }{w - \xi _ {2} } = \mu \frac{z - \xi _ {1} }{ z - \xi _ {2} } , $$

where the multiplier of the mapping is $ \mu > 0 $, $ \mu \neq 1 , \infty $. A unimodular fractional-linear mapping (1) is hyperbolic if and only if $ a + d \in \mathbf R $ and $ | a + d | > 2 $.

2) Each circle of $ \Sigma ^ \prime $ is transformed into itself; such a fractional-linear mapping is said to be elliptic and, in normal form (3), is characterized by a multiplier $ \mu $ such that $ | \mu | = 1 $, $ \mu \neq 1 $. A unimodular fractional-linear mapping (1) is elliptic if and only if $ a + d \in \mathbf R $, $ | a + d | < 2 $.

3) None of the circles of the families $ \Sigma $ and $ \Sigma ^ \prime $ is transformed into itself; such a fractional-linear mapping is said to be loxodromic and, in normal form (3), is characterized by a multiplier $ \mu \in \mathbf C $, $ | \mu | \neq 1 $, such that either $ \mathop{\rm Im} \mu \neq 0 $ or $ \mu < 0 $. A unimodular fractional-linear mapping (1) is loxodromic if and only if $ a + d \in \mathbf C \setminus \mathbf R $.

If two fixed points merge into one point $ \xi _ {1} $, the fractional-linear mapping is said to be parabolic. The family $ \Sigma $ in such a case consists of all the circles having a common tangent at $ \xi _ {1} $; each circle is transformed into itself. The normal form of a parabolic fractional-linear mapping is

$$ \frac{1}{w - \xi _ {1} } = \frac{1}{z - \xi _ {1} } + \alpha ,\ \alpha \in \mathbf C ,\ \alpha \neq 0 , $$

if $ \xi _ {1} \neq \infty $, or

$$ w = z + \alpha ,\ \alpha \in \mathbf C ,\ \alpha \neq 0 , $$

if $ \xi _ {1} = \infty $. A unimodular fractional-linear mapping (1) is parabolic if and only if $ a + d = \pm 2 $.

Owing to the many elementary properties listed above, fractional-linear mappings find extensive use in all branches of the theory of functions of a complex variable and in various applied disciplines. In particular, a model of Lobachevskii geometry can be constructed with the aid of fractional-linear mappings.

Among the subgroups of the complete group of fractional-linear mappings discrete groups $ \Gamma $ of fractional-linear mappings are the most important as regards applications to the analytic theory of differential equations, the theory of automorphic functions and other problems in analysis. Elementary discrete groups of fractional-linear mappings are the finite groups; they are isomorphic either to the cyclic rotation groups of the Riemann sphere or to the rotation groups of regular polyhedra. A discrete group $ \Gamma $ of fractional-linear mappings with an invariant circle $ \gamma $ in $ \overline{\mathbf C}\; $ which is common for all transformations of $ \Gamma $ and for which the interior of $ \gamma $ is transformed into itself under all transformations of $ \Gamma $, is known as a Fuchsian group. A Fuchsian group cannot contain a loxodromic fractional-linear mapping. Historically, the first example of a Fuchsian group was the modular group appearing in the theory of elliptic functions (see also Modular function). The modular group consists of all unimodular fractional-linear mappings (1) in which the coefficients $ a $, $ b $, $ c $, $ d $ are integers; the real axis is invariant with respect to modular fractional-linear mappings. Non-elementary, non-Fuchsian groups of fractional-linear mappings — Kleinian groups (cf. Kleinian group) — are more complicated and have been studied to a lesser extent.

A fractional-linear mapping of the complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, is a non-degenerate mapping

$$ z = ( z _ {1} \dots z _ {n} ) \rightarrow $$

$$ \rightarrow \ w = ( w _ {1} \dots w _ {n} ) = ( L _ {1} ( z) \dots L _ {n} ( z) ) $$

realizable by fractional-linear functions

$$ L _ {k} ( z) = \frac{a _ {1k} z _ {1} + \dots + a _ {nk} z _ {n} + b _ {k} }{c _ {1k} z _ {1} + \dots + c _ {nk} z _ {n} + d _ {k} } , \ k = 1 \dots n . $$

The most important fractional-linear mappings of $ \mathbf C ^ {n} $ are those which extend to some compactification of $ \mathbf C ^ {n} $. Thus, all linear transformations involving a rearrangement of coordinates, as well as fractional-linear mappings of the type

$$ z = ( z _ {1} \dots z _ {n} ) \rightarrow w = ( L _ {1} ( z _ {1} ) \dots L _ {n} ( z _ {n} ) ) , $$

where $ L _ {k} ( z _ {k} ) $ is a fractional-linear mapping of the type (1) in the plane $ z _ {k} $, extend to the function-theoretic space $ \overline{ {\mathbf C ^ {n} }}\; $. The group of fractional-linear mappings generated by these mappings coincides with the group $ \mathop{\rm Aut} \overline{ {\mathbf C ^ {n} }}\; $ of all biholomorphic automorphisms of the compactification $ \overline{ {\mathbf C ^ {n} }}\; $. The corresponding subgroup $ \mathop{\rm Aut} U ^ {n} $, with

$$ L _ {k} ( z _ {k} ) = e ^ {i \theta _ {k} } \frac{z _ {k} - \alpha _ {k} }{1 - \overline \alpha \; _ {k} z _ {k} } ,\ | \alpha _ {k} | < 1 ,\ \mathop{\rm Im} \theta _ {k} = 0 , $$

exhausts all the automorphisms of the unit polydisc $ U ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | < 1, j = 1 \dots n } \} $. Fractional-linear mappings in which

$$ \tag{4 } L _ {k} ( z) = \frac{a _ {1k} z _ {1} + \dots + a _ {nk} z _ {n} + b _ {k} }{c _ {1} z _ {1} + \dots + c _ {n} z _ {n} + d } = \frac{l _ {k} ( z) }{l ( z) } , $$

extend to the projective closure $ \mathbf C P ^ {n} $ of the space $ \mathbf C ^ {n} $. This extension has the following form in homogeneous coordinates:

$$ ( z _ {0} \dots z _ {n} ) \rightarrow \left ( z _ {0} l \left ( \frac{z}{z _ {0} } \right ) \dots z _ {0} l _ {n} \left ( \frac{z}{z _ {0} } \right ) \right ) . $$

These mappings exhaust the group $ \mathop{\rm Aut} \mathbf C P ^ {n} $ of all biholomorphic automorphisms of $ \mathbf C P ^ {n} $. The automorphisms of the unit ball $ B ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z | < 1 } \} $ form the subgroup $ \mathop{\rm Aut} B ^ {n} $ of the group $ \mathop{\rm Aut} \mathbf C P ^ {n} $ consisting of all fractional-linear mappings of the type (4) whose coefficients are subject to certain supplementary conditions (cf. [2], Vol. 2).

References

[1]	I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian)
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[3]	S. Stoilow, "The theory of functions of a complex variable" , 1 , Moscow (1962) (In Russian; translated from Rumanian)
[4]	L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951)

Comments

A good reference for $ \mathop{\rm Aut} B ^ {n} $ is [a1]. Fractional-linear mappings are also known as Möbius transformations.

References

[a1]	W. Rudin, "Function theory in the unit ball in " , Springer (1980)
[a2]	Z. Nehari, "Conformal mapping" , Dover, reprint (1975)
[a3]	H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979)
[a4]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)