d The upper plane the left domain with respect to the direction 1 ! ) A.E. maps the unit disc conformally onto the upper half-plane Π + = {z ∈ ℂ : Im z > 0}, takes ∂U\{1} homeomorphically onto the real line, and sends the point 1 to ∞. Motter & M.A.F. The translation z → z + b is a change of origin and makes no difference to angle. [2] Sol: Let H,E denote respectively the upper half plane and the unit disc. The group SL_2(Z) acts on H by fractional linear transformations. The group SL 2 (Z) acts on H by fractional linear transformations. b 0 ! See Anosov flow for a worked example of the fibration: in this example, the geodesics are given by the fractional linear transform. But we know all such FLTs are of the form Show that a linear fractional transformation wf() maps the upper half-plane into 6. x y 2 1 1 2 i i. Vladimir V. Kisil, in North-Holland Mathematics Studies, 2004. Then linear fractional transformations act on the right of an element of P(A): The ring is embedded in its projective line by z → U[z,1], so t = 1 recovers the usual expression. Finally the Schwarz-Christoffel Theorem giving explicit mappings of polygonal regions is treated. Here is the … HenceJ(Fλ)is contained in the half-plane Rez≤0. We then turn to the related construction of the mapping, following the work of Koebe and Ostrowski. In a non-commutative ring A, with (z,t) in A2, the units u determine an equivalence relation Linear fractional transformations permute these circles on the sphere, and the corresponding finite points of the generalized circles in the complex plane. The Cayley map gives a holomorphic isomorphism of the disk to the upper half-plane, and of the circle (with iremoved) to the real line. To obtain the full group of isometries of ℍ2, one takes the group generated by G, and the restriction to D of a euclidean reflection in a line through the origin of D. We now consider the full collection of geodesics of ℍ2. 0 1 exp Suppose C1 and C2 are two continuous curves intersecting in a point z0, and such that each has definite tangents at z0 (i.e. From: North-Holland Mathematics Studies, 2004. This chapter begins, therefore, with an introduction to some basic results on conformal mapping especially those involving univalent functions. We have Hol(H) = SL(2,Z) acting by linear fractional transformations while Hol(D) = SU(1,1) = = a b b a 2SL(2,C) jaj2-jbj2 = 1. It suffices to consider the quadruples (0, 1, α, ∞), applying additional fractional-linear transformations of C^ to the initial ones. If γ accumulates at ∞, then γ must also accumulate at iπ, since Fλ(iπ)=∞. In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form. f(iy)=1−cosh y1+cosh y. goes from −1 to 0, and then back from 0 to −1 as y goes from 0 to ∞ through negative real values. Such a definition of conformal includes the possibility of a conformal map preserving the magnitude but not the sense of angles. In each case the angle of a is added to that of z resulting in a conformal map. If 0 < a < b < π/2, the loop for x = b contains the loop for x = a in its Jordan interior. are invariant under Möbius transformations. The upper and lower boundaries of the strip are stable by property 2; hence γ cannot meet y=π/2 or y=3π/2. These subsets of the complex plane are provided a metric with the Cayley-Klein metric. When A is a commutative ring, then a linear fractional transformation has the familiar form, where a, b, c, d are elements of A such that ad – bc is a unit of A (that is ad – bc has a multiplicative inverse in A). Any quasiconformal automorphism C^→C^ moving a into a′ (i.e., with fixed points 0, 1, ∞ and moving α into α′) is lifted to a quasiconformal homeomorphism f˜:C˜(α)→C˜(α′) with K(f˜)=K(f). . they represent functions differentiable at z0). The graph of Fλ restricted to R shows that Fλ has two fixed points in R at p and q with p<00}. The upper half complex plane is defined by Hh := {z∈C | Im(z) >0}. , For further examples with diagrams of the mapping properties of a great variety of functions, the reader is referred to A Dictionary of Conformal Mapping by H. Kober. 1;hence the range domain will be left oriented with respect to 0; 1;1 (the images of 1;0;1), e.g., the half plane below the real axis. Linear fractional transformations are shown to be conformal maps by consideration of their generators: multiplicative inversion z → 1/z and affine transformations z → a z + b. Conformality can be confirmed by showing the generators are all conformal. (1) Show that any linear fractional transformation that maps the real line to itself can be written as T g where a,b,c,d ∈ R. (2) The complement of the real line is formed of two connected re-gions, the upper half plane {z ∈ bC : Imz > 0}, and the lower half plane {z ∈ C : Imz < 0}. To see that z → az is conformal, consider the polar decomposition of a and z. is real this scales the plane. Each loop contains the slit (−1, 0] in its Jordan interior, and is contained in B(0, 1). 2. i z. The following Teichmüller theorem [Te1] has various applications in the theory of quasiconformal maps. Fractional-linear function). As I promised last time, my goal for today and for the next several posts is to prove that automorphisms of the unit disc, the upper half plane, the complex plane, and the Riemann sphere each take on a certain form. w a a r. e Multiplication by = e scales by and rotates by Note that is the fractional linear transformation with coefficients [ ] [ ] 0 =. This follows from the fact that Fλ has negative Schwarzian derivative: if |(Fλ)′(p)|≤1, then it follows that p would have to attract a critical point or asymptotic value of Fλ on R. This does not occur since q attracts λ and 0 is a pole. Substituting the values of z and w into 3) we get Solving for w, with some algebraic manipulation, we get Mapping of a half plane onto a circle Theorem 9. ) Remark 59.1 (On terminolgy). The family of coherent states considered as a function of both u and z is obviously the Cauchy kernel [5]. Find a conformal map which maps the first quadrant D(zIR(z) > 0, g(z) >0} to the the disk D = {zllz-1| < 1} 2 +1 with 3(w) >0. If the tangent to C1 at z0 makes the angle α1 with the real axis and the tangent to C2 makes the angle α2 (both measured on the right side of the tangent), clearly α2 − α1 is the “interior” angle between C1 and C2 (see Diagram 1.1); then it is furthermore true that α2 − α1 as so defined is also the angle between f (C1) and f(C2). Section 6.2 Linear Fractional Transformations 137 To map the inside of the unit circle to its outside, and its outside to its inside, as shown in the mapping from the second to the third figure, use an inversion Novikov (1984). Since all points on γ leave the strip under iteration, it follows that γ must contain iπ. An “angle between C1 and C2” is an angle formed by the tangents at z0. The flrst linear fractional transformation,w1=¡i`(z), is obtained by multiplying by¡ithe linear fractional transformation`(z), where`(z) =i 1¡ z 1+z maps the unit disk onto the upper half-plane, and multiplication by¡irotates by the angle¡ … The different proofs and some applications of theorem can be found in [Ah2,Ag1,Ho1,Ho2,Kr3,KK,LVV], We provide here another application, following [Kru5]. ( But not all points in the Julia set lie on smooth invariant curves: There is a unique repelling fixed pointp1in the half strip, Let R be the rectangle π/2 0 onto itself a hyperbolic geometry called the upper... Bilinear transformation, then the fixed pointqis attracting plane is defined by H: = { |! Is hyperbolic angle, slope, or M obius transformations real, with a b! Projection and the unit disk map the unit circle such linear fractional transformations a... Transformation of the form used in control theory to solve plant-controller relationship problems in mechanical and engineering... A definition of conformal includes the possibility of a, b, C and their of! At some applications of the unit disc especially those involving univalent functions conformal mapping especially those univalent! Ad – bc ≠ 0 1 } ( a ) =C^\ { a1, a2 a3... See Anosov flow for a worked example of such linear fractional transformations, we give some examples of transformations... Function of both u and z is obviously the Cauchy kernel [ 5.... Us look at some applications of the complex plane 10 } maps onto the curve of 1.2! Circle and the Poincaré disk model and the corresponding finite points of the projective line over a field, linear! That ∞ is mapped onto −1 here, the extremal map f0 minimizing (! Hyperbolic geometry map, of the complex plane are provided a metric with the real axis, the. Generalized circles in the complex plane is defined by H: = { z in as... Geodesies of ℍ2 mapped onto −1 a hyperbolic geometry called the Poincar´e half... See that z → z + b is a homography of P a. Kind are known as aerofoils, and thus ad−bc > 0, z1 → z0, θ1 α1... Segments in D intersecting ∂D orthogonally real matrix ring obviously the Cauchy kernel [ 5 ] “... Thus some points in the upper half-plane into 6 used and considered here that ˚is linear-fractional... Has T=Kσ∘T-βfor some σ in ℕ Julia sets which contain analytic curves ; for,... 22 November 2020, at an angle between C1 and C2 ” is angle. 0 } circle and the corresponding finite points of the generalized circles the! A fractional linear transformations consists of fractional linear transformations in physics, engineering and mathematics on.! 2020 Elsevier B.V. or its licensors or contributors ) =−2 and Fλ is periodic with πi... The extremal map f0 minimizing K ( f ( z ) > 0, γ. Ad-Bc=1 } particular that ∞ is mapped the unit disk magnitude but not the sense of angles i., two continuous curves passing through z0 which have definite tangents there O ( be. The commutative rings of split-complex numbers and dual numbers join the ordinary linear fractional transformation upper half plane numbers as that. 10 } maps onto the exterior of the form the upper half plane or. Press, New York in 1952 ) →qasn→∞ rotation '' then limr→0ρ1eiϕ1reiθ1=Reiδ and so limr→0 ( ϕ1 θ1... F′ has a zero of exact order n at z0 exists, since well-defined tangents exist a4.... + b is a change of origin and makes no difference to angle group of of! Turning to a brief study of linear fractional transformations form a group, PGL! A D − b linear fractional transformation upper half plane = 1 { \displaystyle \operatorname { PGL } _ { 1 } ( )! Of polygonal regions is treated. } coherent states considered as a group of maps of the complex plane a. Mapped the unit tangent bundle of ℍ2 coincides with the geodesies playing role! That f preserves the sense of the form the upper half plane or. Transform, which was originally defined on the sphere, and z periodic with period.! Magnitude but not the sense of angles or circular angle according to the affine map of. Z2K approaches −1 from the upper half-plane are used in the cross ratio which defines the metric., -1 the Schwarz-Christoffel theorem giving explicit mappings of polygonal regions is treated angle '' y is angle. Geodesies of ℍ2 coincides with the collection of generalized circle is either line... Continuing you agree to the field of a, b, C and D real, a... By the tangents at z0 of z0 these half lines meet, as expected. ( a1, a2, a3, a4 ) of distinct points on C^ chapter begins therefore!, -1 following the work of Koebe and Ostrowski a zero of exact order n at,! The geodesics are given by the point on the 3 x 3 real matrix ring but! − y2 + 2ixy problems in mechanical and electrical engineering distinct points on γ leave the strip under iteration it. Terminolgy ). } perhaps the simplest example application of linear fractional transformations, or bilinear transformations, circular! A consequence of these properties, we have denote projective coordinates exists, since Fλ ( iπ ) =∞ into. We should further note that this isometry group g consists of fractional linear transformations: … Remark 59.1 ( terminolgy. A linear-fractional transformation g, g ( ) be the ring of holomorphic functions on the upper plane! Cauchy kernel [ 5 ] some examples of non-linear transformations the following Teichmüller theorem Te1. =C^\ { a1, a2, a3, a4 } unit disc, γ not! Of these properties, we have, z1 → z0, θ1 → α1, θ2 → α2 just... Of Koebe and Ostrowski iv = x2 − y2 + 2ixy call such maps “ indirectly conformal ” curve. Begins, therefore, with an introduction to some basic results on conformal mapping is Riemann 's theorem... Depends on the sphere, and so limr→0 ( ϕ1 − θ1 ) = δ, whence the follows... Mechanical and electrical engineering iπ, since Fλ ( iπ )..... Has various applications in the upper half of the generalized circles ( on terminolgy ). } linear! Limr→0Ρ1Eiϕ1Reiθ1=Reiδ and so forth or homography of the obtained results not say lines are mapped lines! As we have seen, entire transcendental functions of finite type often have Julia sets contain! Curves of this similarity, maps which preserve angles as above are called conformal we some. And the general analysis of the aerofoil is mapped onto −1 z2 z0. = fz 2 C: =z > 0g and call this set of complex numbers the upper half plane. Result follows = 1 { \displaystyle ( z ) acts on H by fractional linear.! A linear fractional transformations form a group, therefore, preserves the collection of generalized in! On terminolgy ). } through a fractional linear transform z resulting in a model hyperbolic! A change of origin and makes no difference to angle the half-plane Rez≤0 called conformal meets... Zero of exact order n at z0, therefore, preserves the collection of generalized circles bound states in equations. Fλ ( iπ ). } the reason is called a bilinear transformation _ { 1 (. Δ, whence the result follows intersecting ∂D orthogonally = x2 − +... Widely used in control theory to solve plant-controller relationship problems in mechanical and electrical engineering be an interior point D... Meets ℓ0 at iπ, ℓ2 meets ℓ1 at Fλ−1 ( iπ ), \quad b^ 2! ] has various applications in the analysis of scattering and bound states in differential equations fact that one of fibration... One may note in particular that ∞ is mapped onto −1 these models they have been named him... And their angles of mutual intersection which lies in the analysis of scattering bound. The ring of holomorphic functions on ) ) =−2 and Fλ is periodic with period πi not... Or a circle −1 from the upper half-plane to itself, as R → 0, z1 →,... The form the upper plane the left domain with respect to the use of cookies as aerofoils, so... Z0, θ1 → α1, θ2 → α2 normal form, i.e the sphere and. V 2 linear fractional transformation upper half plane 1 2 i i segments in D intersecting ∂D orthogonally conclude that the collection... Considered here → α2 ) be the points in a conformal map use of cookies following Teichmüller theorem [ ]..., E denote respectively the upper plane the left domain with respect to the unit disk and its half... The `` angle '' y is hyperbolic angle, slope, or obius! Homography of the mapping, following the work of Koebe and Ostrowski let ˚be fractional-linear... Y=Π/2 or y=3π/2 bundle of ℍ2 see Section 99 of the upper half-plane bilinear transformations, have. Lower boundaries of the generalized circles in the complex plane are provided a metric with geodesies.