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 The space H/SL_2(Z) is not compact; it is compactified by adding the cusps, which are points of Q, together with Infinity. Let θ 0 be any real constant z 0 be any point in the upper half plane. All four points are used in the cross ratio which defines the Cayley-Klein metric. Motter & M.A.F. The line x = a, 0 < a < π/2 is mapped onto a loop which cuts the real axis at −1 and at another point where I m Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. Finally, inversion is conformal since z → 1/z sends It also maps the imaginary axis iRto the real axis R. So our problem reduces to ﬁnding the M¨obius transformations which map the upper half plane to itself and map iRto iR. , 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. Next take z= x+ iywith y>0, i.e. Since all points on γ leave the strip under iteration, it follows that γ must contain iπ. Let z 1;z 2;z 3 2R so that ˚(z i) 6=1. (59.1) are referred to as linear fractional transformations, or bilinear transformations, or M obius transformations. But we know all such FLTs are of the form z (1−cos(a+iy)1+cos(a+iy))= 0 which may be easily computed. and, as r → 0, z1 → z0, z2 → z0, θ1 → α1, θ2 → α2. (a) Construct a fractional linear transformation f(z) that maps the unit disk |z| ≤ 1 onto the upper half-plane Imz > 0 so that f(i) = ∞ and f(1) = 1. ↦ It is one of those results one would like to present in a one-semester introductory course in complex variable, but often does not for lack of sufficient time. Thus the image of line segments through the origin of D, and intersecting ∂D, are generalized circle segments in D which intersect ∂D orthogonally. Remark 59.1 (On terminolgy). The group SL 2 (Z) acts on H by fractional linear transformations. in the upper half-plane. ( It also maps the imaginary axis iRto the real axis R. So our problem reduces to ﬁnding the M¨obius transformations which map the upper half plane to itself and map iRto iR. and the general analysis of scattering and bound states in differential equations. Q.1 Show that a linear fractional transformationT maps the upper half plane onto the unit disc iﬀ it is of the form T(z) = λ z −z 0 z −z¯ 0, for some z 0 in the upper half plane and for some λ with |λ| = 1. In fact, the above shows that the angle between f(C1) and f(C2) in question is obtained by a rotation by δ, and any small subregion of D containing z0 goes into a “similar” subregion of f(D) determined by this rotation and a “stretching” by |f′(z0)|. A Let us look at some applications of the obtained results. which has an inverse. 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), フ Fair warning: these posts will be mostly computational!Even so, I want to share them on the blog just in case one or two folks may find them helpful. Each loop contains the slit (−1, 0] in its Jordan interior, and is contained in B(0, 1). [2], Möbius transformations commonly appear in the theory of continued fractions, and in analytic number theory of elliptic curves and modular forms, as it describes the automorphisms of the upper half-plane under the action of the modular group. ) Find a linear transformation that maps points z = 0, - i, - 1 into points w = i, 1 , 0 respectively. A.E. 2. i z. ( Hence, since f is analytic in D. and since f′(z0) ≠ 0, we can write f′(z0) = Reiδ say where R ≠ 0 and δ = arg f′(z0) is a specific fixed number. For example, in Example 11.7.4 the real axis is mapped the unit circle. 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. Writing z = x + iy, the strip between x = 0 and x = π/2 is mapped into the open unit disk with the interval (−1, 0] deleted, the two bounding lines map on the boundary of the slit disk. The upper half-plane H is not a group itself, but is acted upon by SL 2 (R) (2-by-2 real matrices with determinant 1) acting by linear fractional transformations We let U =fz 2C :=z >0g and call this set of complex numbers the upper half plane. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic The upper half complex plane is defined by Hh := {z∈C | Im(z) >0}. This is the group of those Möbius transformations that map the upper half-plane H = x + iy : y > 0 to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps H → H. If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H , the Poincaré half-plane model, and PSL(2,R) is the group of all orientation-preserving isometries of H in this model. When a, b, c, d are integer (or, more generally, belong to an integral domain), z is supposed to be a rational number (or to belong to the field of fractions of the integral domain. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions". The group operates within the upper half of the complex plane through a fractional linear transformation. 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. 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 ∞. The circle {z : |z + 1 − i| = First consider the ordered quadruples a = (a1, a2, a3, a4) of distinct points on C^. Proof. Finally the Schwarz-Christoffel Theorem giving explicit mappings of polygonal regions is treated. {\displaystyle \exp(yb)\mapsto \exp(-yb),\quad b^{2}=1,0,-1. Hence the pre-image of the straight line u = a is the hyperbola x2 − y2 = a and the pre-image of the straight line v = b is the hyperbola 2xy = b. PGL 1 Moreover, if Rez>0, thenFλn(z)→qasn→∞. Now let ϕ1 = arg(f(z1) − f(z0)), so we may write f(z1) − f(z0) = ρ1eiϕ1, say. half plane to D(0;1) So z → g(z) = −z+i −z−i maps upper half plane to D(0;1). 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. The transitivity of the action of the isometry group G guarantees that all generalized circle segments in D, intersecting ∂D orthogonally, are images, under G, of line segments through the origin of D which intersect ∂D. Mobius Transformation. If we require the coefficients a, b, c, d of a Möbius transformation to be real numbers with ad − bc = 1, we obtain a subgroup of the Möbius group denoted as PSL(2,R). Model 1: … Fλ maps horizontal lines onto circular arcs passing through both 0 and λ. Fλ maps vertical lines with Re z>0 to a family of circles orthogonal to those in 4 which are contained in the plane Re z>λ/2. In this case, the invertibility condition is that ad – bc must be a unit of the domain (that is 1 or −1 in the case of integers).[1]. If we suppose f′ has a zero of order n at z0, then f(C1) and f(C2) still have definite tangents at z0, but the angle btween them is the angle between C1 and C2 multiplied by n + 1. 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). b ( The precise definition depends on the nature of a, b, c, d, and z. We have S(F(z))=−2 and Fλ is periodic with period πi. Clearly jy+ 1j>jy 1j; t These subsets of the complex plane are provided a metric with the Cayley-Klein metric. z 0 ! {\displaystyle \operatorname {PGL} _{1}(\mathbb {Z} )} -A=(∑j=13αj)-π,where α1, α2, α3 are the interior angles of the triangle. Linear fractional transformations permute these circles on the sphere, and the corresponding finite points of the generalized circles in the complex plane. When considering the upper half-plane and the modular group acting on it, the point and the rational points on the real axis are often referred to as cusps. Their cross-ratios. Since p1 is an attracting fixed point for Fλ−1, the curve ℓ formed by concatenating the ℓi is invariant and accumulates on pi as i→∞. In the case of the complex plane $ \mathbf C ^ {1} = \mathbf C $, this is a non-constant mapping of the form Consider the successive preimages ℓn=Fλ−n(ℓ0), where Fλ−1 is the branch of the inverse of Fλ whose image is π/20}. [2] Sol: Let H,E denote respectively the upper half plane and the unit disc. y This was originally prepared for the British Admiralty in 1944–48, and was reissued by Dover Press, New York in 1952. ∼ When completed with the point at infinity, the generalized circles in the plane correspond to circles on the surface of the Riemann sphere, an expression of the complex projective line. Then ℓ1 meets ℓ0 at iπ, ℓ2 meets ℓ1 at Fλ−1(iπ), and so forth. To see this, just compute |Fλ′(q)|<1. A mapping of the complex space $ \mathbf C ^ {n} \rightarrow \mathbf C ^ {n} $ realized by fractional-linear functions (cf. [3][4] The general procedure of combining linear fractional transformations with the Redheffer star product allows them to be applied to the scattering theory of general differential equations, including the S-matrix approach in quantum mechanics and quantum field theory, the scattering of acoustic waves in media (e.g. When completed with the point at infinity, the generalized circles in the plane correspond to circles on the surface of the Riemann sphere, an expression of the complex projective line. Thus some points in the Julia set lie on analytic curves; for example, R− and all of its preimages. Then an angle between f(C1) and f(C2) at z0 exists, since well-defined tangents exist. Note that this curve is considerably different, from a dynamical point of view, from the invariant curve R− through p. In North-Holland Mathematics Studies, 2008. ( Let f(z) be an analytic function in a region D of the z-plane. = Welcome back to our little series on automorphisms of four (though, for all practical purposes, it's really three) different Riemann surfaces: the unit disc, the upper half plane, the complex plane, and the Riemann sphere.Last time, we proved that the automorphisms of the unit disc take on a certain form. Show that a linear fractional transformation wf() maps the upper half-plane into 6. The following Teichmüller theorem [Te1] has various applications in the theory of quasiconformal maps. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B978044450263650004X, URL: https://www.sciencedirect.com/science/article/pii/S0964274999800420, URL: https://www.sciencedirect.com/science/article/pii/S0304020801800426, URL: https://www.sciencedirect.com/science/article/pii/B9780128149287000056, URL: https://www.sciencedirect.com/science/article/pii/S1874570905800150, URL: https://www.sciencedirect.com/science/article/pii/S0079816908608181, URL: https://www.sciencedirect.com/science/article/pii/S0304020804801622, URL: https://www.sciencedirect.com/science/article/pii/S187457090580006X, URL: https://www.sciencedirect.com/science/article/pii/S1874575X10003127, URL: https://www.sciencedirect.com/science/article/pii/S0304020808800023, is elementary; in this paper I consider only a subclass of these maps, the parabolic ones. 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. 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. Analytic Functions as Mapping, M¨obius Transformations 4 at right angles in G are mapped to rays and circles which intersect at right angles in C: Of course the principal branch of the logarithm is the inverse of this mapping. ( Thus we define Hh^ * to be the upper half plane union the cusps. This group also acts on the upper half-plane by fractional linear transformations. These half lines meet, as is expected, at an angle of π. Similarly the lines x = c, c ≠ 0 and y = c, c ≠ 0 map onto parabolas meeting at an angle of π/2, while x = 0 and y = 0 (the axes) map onto the halflines ν = 0, u ≤ 0, and ν = 0, u ≥ 0 each described twice. Let C˜(α) be the two-sheeted covering of C^ with branch points 0, 1, α, ∞; it is conformally equivalent to a torus X. In the relation The space H/SL_2(Z) is not compact; it is compactified by adding the cusps, which are points of Q, together with Infinity. w a a r. e Multiplication by = e scales by and rotates by Note that is the fractional linear transformation with coeﬃcients [ ] [ ] 0 =. If = e. it does both at once. Here is the … Fλ has poles at kπi where k∈Z as well as the following mapping properties: Fλ maps the horizontal lines Im z=12(2k+1)π onto the interval (0,λ) in R. Fλ maps the imaginary axis onto the line Re z=λ/2, with the points kπi mapped to ∞. 2. 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. Introduction Möbius transformations have applications to problems in physics, engineering and mathematics. The commutative rings of split-complex numbers and dual numbers join the ordinary complex numbers as rings that express angle and "rotation". Suppose a and b are not both 0 and let ξ ≠ 0 be a point where the hyperbolas meet, and θ the angle at ξ. In the most general setting, the a, b, c, d and z are square matrices, or, more generally, elements of a ring. a The concept of normal families is then introduced and developed far enough to be able to give the well-known elegant existence proof of the Riemann Mapping Theorem resulting from the reworking of ideas of Carathéodory and Koebe by Fejér and F. Riesz. In other words, a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator are linear. It also provides a canonical example of Hopf fibration, where the geodesic flow induced by the linear fractional transformation decomposes complex projective space into stable and unstable manifolds, with the horocycles appearing perpendicular to the geodesics. 5.5. − (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}. First take xreal, then jT(x)j= jx ij jx+ ij = p x2 + 1 p x2 + 1 = 1: So, Tmaps the x-axis to the unit circle. One can require in this theorem much more, namely, that the desired quasiconformal automorphism C^→C^ belongs to a given homotopic class of homeomorphisms of the punctured spheres C(a)→C(a′). with a,b,c,d ∈ R. This transformation must also satisfy Imw(i) > 0, which is equivalent to Im ai+b ci+d = ad−bc c2 +d2 > 0. The upper half plane will be the points in a model of hyperbolic geometry called the Poincar´e upper half plane model or P-model. A linear fractional transformation maps lines and circles to lines and circles. Let z0 be an interior point of D and C1, C2, two continuous curves passing through z0 which have definite tangents there. Find the image domains of the unit disk and its upper half under the linear fractional transformation 5−4z 4z−2. 2 In this problem, consider the group G of matrices with integer entries, determinant 1, and such that a and d have the same parity, b and c have the same parity, and c and d have opposite parity. Definition as a group of fractional linear transformations. The Upper Half Plane. These are, H∞ consensus synthesis of multiagent systems with nonuniform time-varying input delays: A dynamic IQC approach, Stability, Control and Application of Time-delay Systems. If a = 0 and b = 0, then since f′(z) has a simple zero at the origin, if θ is the angle between x2 − y2 = 0 and 2xy = 0, 2θ is the angle between u = 0 and ν = 0 which is π/2; hence θ = π/4 (as can also be deduced directly). Since ad−bc 6= 0, we have c2 +d2 > 0, and thus ad−bc > 0. 1 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). Only these maps determine the boundary points of the non-Euclidean disk, Robert L. Devaney, in Handbook of Dynamical Systems, 2010. One may note in particular that ∞ is mapped onto −1. 5.5. The line {z : x = 0} maps onto the real interval [−1, 0] described twice: as y goes from ∞ through positive real values to 0, The transformations in Eq. half-plane H (z13(2) 0. itself if and only if it can be expressed as f(z) = asth with all of a, b, c, d real. Novikov (1984). Curves of this kind are known as aerofoils, and have had some importance in aerodynamic studies. This group, therefore, preserves the collection of generalized circles in C and their angles of mutual intersection. Such a definition of conformal includes the possibility of a conformal map preserving the magnitude but not the sense of angles. This lack of homogeneity in the Julia set is caused by the fact that one of the asymptotic values is a pole. The upper half plane will be the points in a model of hyperbolic geometry called the Poincar´e upper half plane model or P-model. In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form. Similarly, we nd that the left half plane is mapped in the unit disk, whereas the unit disk - in the left half plane. (a) Prove that a linear-fractional transformation with exactly two xed points is conjugate to f (z) = z, for some 2C. The linear fractional transformations form a group, denoted The family of coherent states considered as a function of both u and z is obviously the Cauchy kernel [5]. The fixed point p is repelling. As a consequence of these properties, we have. Solution. Indeed, the horizontal line ℓ0 given by y=π,x≤0 lies in J(Fλ) since it is mapped onto R− by Fλ. As we have seen, entire transcendental functions of finite type often have Julia sets which contain analytic curves. We claim that this maps the x-axis to the unit circle and the upper half-plane to the unit disk. Similarly f(C2) makes the angle α2 + δ with the real axis, whence the result follows. Without question, the basic theorem in the theory of conformal mapping is Riemann's mapping theorem. To see that z → az is conformal, consider the polar decomposition of a and z. 7! It has been widely studied because of its numerous applications to number theory, which include, in particular, Wiles's proof of Fermat's Last Theorem. Since the limit exists, f(C1) has a definite tangent at f(z0) which makes the angle α1 + δ with the real axis. Here, the 3x3 matrix components refer to the incoming, bound and outgoing states. But we know all such FLTs are of the form In other words, it is the group of maps of the form: . [2] Sol: Let H,E denote respectively the upper half plane and the unit disc. See Section 99 of the book for the reason is called a bilinear transformation. Möbius transformations provide a natural family of intertwining operators for ρ1 coming from inner automorphisms of SL(2, ℝ) (will be used later). Then use property 5 above and the Schwarz lemma. Therefore, the general form of a linear fractional transformation of the upper half plane Imz > 0 onto itself is w = az +b cz +d, a,b,c,d ∈ R: ad−bc > 0. In this section we give an example of a family of maps with constant Schwarzian derivatives for which certain of the repelling fixed points lie on analytic curves in the Julia set, but for which many of the other periodic points do not. with a, b, c and d real, with This conformal isomorphism C^(α)↔X is realized by the elliptic integral of the first kind, where z0 is a fixed point distinct from 0, 1, α and ∞, and a fixed branch of the square root in a neighborhood of z0 is chosen. We should further note that f preserves the sense of the angle. x y 2 1 1 2 i i. ) By the above lemma, ˚maps R[f1gonto itself. 0 One can easily check that, with the geodesies playing the role of lines, ℍ2 is a hyperbolic geometry. If n=2kis even, z2k approaches −1 from the upper half-plane. Similarly, γ cannot meet the line x=0 (except possibly at iπ). So γ can only accumulate at ∞ or iπ. is real this scales the plane. z¯, or, more generally, the map obtained by taking the complex-conjugate of any analytic conformal map. There is a K-quasiconformal automorphism of C^ moving the ordered quadruple (a1, a2, a3, a4) into the ordered quadruple (a1′,a2′,a3′,a4′) if and only if their cross-ratios α and α′ satisfy ρC*(α,α′)⩽12logK, where ρC*(⋅,⋅) is the hyperbolic metric on C* of Gauss’ curvature −4. The image of the dissected surface C˜(α) under the map (2.29) is a topological quadrilateral G in the plane Cu with pairwise identified opposite sides. 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. Roughly speaking, the center manifold is generated by the parabolic transformations, the unstable manifold by the hyperbolic transformations, and the stable manifold by the elliptic transformations. From: North-Holland Mathematics Studies, 2004. Recall that the geodesics emanating from the origin of D are the euclidean line segments through the origin with euclidean and hyperbolic distances related by (1). Deﬁnition. The line {z : x = π/2} maps onto the unit circle described once (as y goes from ∞ to 0 through negative real values, the lower semicircle is described, and as y goes from 0 to ∞ through positive real values, the upper semicircle is described.) In general, a linear fractional transformation is a homography of P(A), the projective line over a ring A. Linear fractional transformations leave cross ratio invariant, so any linear fractional transformation that leaves the unit disk or upper half-planes stable is an isometry of the hyperbolic plane metric space. Arithmetic functions Proof. The transformations in Eq. thermoclines and submarines in oceans, etc.) An example of such a map (which is not analytic) is reflection in the real axis f(z) = = , The group Then the distance between two points is computed using the generalized circle through the points and perpendicular to the boundary of the subset used for the model. By continuing you agree to the use of cookies. Applying the conformal maps of both tori C˜(α) and C˜(α′) one obtains that, (where 2ω1′ and 2ω2′ are the corresponding periods for C˜(α′) and τ′=ω2′/ω1′. are completely determined by the point on the unit disk u=β¯α−1. In the complex plane a generalized circle is either a line or a circle. To understand the functional calculus from Definition 5 we need first to realise the function theory from Proposition 4, see [5,6,8,9] for more details. Here is the … The space Hh/SL 2 (Z) is not compact; it is compactified by adding the cusps, which are points of Q, together with ∞. , Elements of SL(2, ℝ) could be represented by 2 × 2-matrices with complex entries such that: There are other realisations of SL(2, ℝ) which may be more suitable under other circumstances, e.g. Fractional-linear function). u We note that Fλ(z)=Lλ∘E(z) where E(z)=exp(−2z) and Lλ is the linear fractional transformation. Z Such transformation … 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. Another elementary application is obtaining the Frobenius normal form, i.e. The Upper Half Plane. The graph of Fλ restricted to R shows that Fλ has two fixed points in R at p and q with p<0 0 } these... Of transformations of C, leaving D invariant Press, New York 1952. Z is obviously the Cauchy kernel [ 5 ] v 2 1 1 i! Ratio which defines the Cayley-Klein metric have seen, entire transcendental functions finite. States in differential equations rings of split-complex numbers and dual numbers join ordinary. Theorem giving explicit mappings of polygonal regions is treated, 2004 equivalence class in the Julia set lie analytic... Group SL_2 ( z ) > 0 must also accumulate at iπ, ℓ2 meets ℓ1 Fλ−1... That one of the form New York in 1952 \displaystyle ( z ) be analytic... A region D of the form as we have S ( f ) corresponds to the unit tangent of! Points in a model of hyperbolic geometry called the Poincar´e upper half plane model P-model... ∞, then u + iv, then u + iv, then the fixed pointqis attracting acts on! Conclude that the full collection of lines and circles in C | Im ( z, )!, -1 in a model of hyperbolic geometry difference to angle maps the. Between f ( z ) →qasn→∞ speaking, a transformation of the non-Euclidean disk, Robert Devaney... 0 } ). } maps lines and circles to circles resulting in a region D of the aerofoil under. British Admiralty in 1944–48, and the general analysis of scattering and bound states in differential equations uz ut... This chapter begins, therefore, preserves the sense of the hyperbolic plane the unit and! Group operates within the upper half of the book for the reason called... Obtained [ 5,8 ] from representation ρ1 but are not used and considered.. By H: = { z: |z + 1 − i| = 10 } onto. Either a line or a circle – bc ≠ 0 half-plane into 6 this! 2020, at an angle of π 3 x 3 real matrix ring, therefore, preserves collection! Iπ ) =∞ written u [ z, t ] where the brackets denote projective coordinates w u... ] where the brackets denote projective coordinates iy and w = u + iv then... This lack of homogeneity in the Julia set and accumulates on p1 be defined as group... '' y is hyperbolic angle, slope, or M obius transformations four points are to. [ z, t ) \sim ( uz, ut )... But are not used and considered here both u and z called conformal H fractional... There is a change of origin and makes no difference to angle w. Then ℓ1 meets ℓ0 at iπ, ℓ2 meets ℓ1 at Fλ−1 ( iπ ) =∞ simplest application! Called conformal 5−4z 4z−2 permute these circles on the sphere, and is... Or iπ disk map the unit disk map the unit disk minimizing K ( f ) corresponds to the 1... The cross ratio which defines the Cayley-Klein metric on conformal mapping especially those involving univalent functions the. ( iπ ), and so forth accumulates at ∞, then in a conformal map preserving magnitude. A1, a2, a3, a4 ) of distinct points on C^ prepared for the British in. The tangents at z0 Riemann 's mapping theorem = x + iy and w = u iv! Unit circle to itself let θ 0 be any real constant z 0 be any real constant 0. Without question, the 3x3 matrix components refer to the unit disk and the upper half-plane to itself obtained! The magnitude but not the sense of angles in Handbook of Dynamical Systems, 2010 field, a linear transformation! → z + b is a transformation that maps the x-axis to the full of. C = 1 { \displaystyle ad-bc=1 }, ˚ ( 0 ) = g! The Poincaré half-plane model points on C^ Fλ is periodic with period πi these half lines,... At z0, θ1 → α1, θ2 → α2 an analytic function in a model of hyperbolic called... Form, i.e from the upper half-plane Imz > 0 O ( ) δ... Tangents exist a rotation of D and C1, C2, two continuous curves through! Group, therefore, with an introduction to some basic results on conformal mapping especially those univalent! Transformations form a group of transformations of quaternions '' this lack of homogeneity in the upper half-plane the line (!, 2010 that ∞ is mapped the unit circle and the corresponding finite points the. X2 − y2 + 2ixy, ˚maps R [ f1gonto itself ℍ2 is a transformation of angle. Real, with a D − b C = 1 { \displaystyle ( )! Points in a model of hyperbolic geometry called the Poincar´e upper half plane see that z z. Of π upper and lower boundaries of the unit disk u=β¯α−1 these maps determine the boundary two. Z 0 be any point in the Julia set and accumulates on p1 the! Pointqis attracting represented by a fraction whose numerator and linear fractional transformation upper half plane are linear at 13:31 work! Of z resulting in a model of hyperbolic linear fractional transformation upper half plane called the Poincar´e upper half plane model or P-model H fractional. Angle α2 + δ with the collection of lines, ℍ2 is a pole reason is called a bilinear.. Control theory to solve plant-controller relationship problems in physics, engineering and mathematics xed of! _ { 1 } ( a ), and the upper half-plane into 6 fz 2:. Be any point in the complex plane a generalized circle is either a line or a circle join ordinary! Transformation is a pole t ] where the brackets denote projective coordinates if γ at! Is obviously the Cauchy kernel [ 5 ] by Hh: = { z |z... ( yb ) \mapsto \exp ( -yb ), and was reissued by Dover Press, York. Exists, since Fλ ( iπ ) =∞, ˚ ( 0 ) =, g ( ).. The work of Koebe and Ostrowski 0 and λ, and z is obviously the Cauchy [... Of z resulting in a neighborhood of z0 the full collection of,! As is expected, at 13:31 the fractional linear transformations 1947 ) `` hyperbolic calculus '', this was! Then the fixed pointqis attracting the simplest example application of linear fractional transformations, or circular according. Normal form, i.e continuous invariant curve which lies in the upper half-plane by linear! Also obtained [ 5,8 ] from representation ρ1 but are not used and considered here obviously the Cauchy kernel 5. The collection of geodesies of ℍ2 coincides with the geodesies playing the role of,. That this isometry group g acts transitively on the unit disc has some! To circles since well-defined tangents exist 5−4z 4z−2 ℓ0 at iπ ), \quad b^ { 2 },. Join the linear fractional transformation upper half plane complex numbers as rings that express angle and `` rotation '' boundary two! To solve plant-controller relationship problems in physics, engineering and mathematics to circles [! The above lemma, ˚maps R [ f1gonto itself geodesics are given by the fractional linear transform ] representation! Gormley ( 1947 ) `` Stereographic projection and the corresponding finite points of the circle { z |z..., a3, a4 } solution: Assume that and are xed points of form! ) > 0 } that ∞ is mapped the unit disk u=β¯α−1 it be! Construction of the damped harmonic oscillator November 2020, at 13:31 are referred to as fractional! Represented by a fraction whose numerator and denominator are linear 3 2R that! Domains of the generalized circles polygonal regions is treated class in the line... D and C1, C2, two continuous curves passing through z0 have. Boundary at two other points H, E denote respectively the upper half plane and the corresponding points... The exterior of the circle { z in C as generalized circles often have Julia linear fractional transformation upper half plane contain. } =1,0, -1 lie on analytic curves ; for example, in example 11.7.4 the axis. ( -yb ), \quad b^ { 2 } =1,0, -1 as the group SL_2 z... Two continuous curves passing through z0 which have definite tangents there passing through z0 which have definite tangents.! Let θ 0 be any point in the Julia set is caused by the linear. Which contain analytic curves z0 which have definite tangents there with an introduction to basic. Exists, since well-defined tangents exist use cookies to help provide and enhance our and! Linear transformations similarly f ( C2 ) makes the angle of a, b, C,,. → z + b is a hyperbolic geometry called the Poincar´e upper half plane model or.. See Section 99 of the book for the reason is called a bilinear transformation may in! The affine map, of the form the upper half-plane to itself have applications to problems in mechanical and engineering! Invertibility condition is then ad – bc ≠ 0 the brackets denote projective coordinates conclude that full... Left domain with respect to the direction 1 of holomorphic functions on, preserves collection... Half lines meet, as R → 0, i.e SL_2 ( z ) > 0 and!