DOCSLIB.ORG

Lecture 15 – the Riemann Mapping Theorem 1 Normal Families

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 15 – the Riemann Mapping Theorem 1 Normal Families Lecture 15 { The Riemann mapping theorem MATH-GA 2451.001 Complex Variables The point of this lecture is to prove that the unit disk can be mapped conformally onto any simply connected open set in the plane, other than the plane itself. We will then know that any two such simply connected open set can be mapped conformally onto each other, since we can map either of these sets to the unit disk. The result above is known as the Riemann mapping theorem. We will prove it using basic theory of normal families. We start this lecture with that. 1 Normal families 1.1 Definition Let Ω be an open connected set in C and F a family of analytic functions on Ω. F is said to be (i) normal or relatively compact if every sequence from F has a subsequence that converges uniformly on every compact subset of Ω (ii) locally uniformly bounded if for any compact subset K of Ω, there exists M > 0 such that jf(z)j ≤ M 8f 2 F , 8z 2 K. (iii) equicontinuous on a compact set K of Ω if 8 > 0 ; 9δ > 0 such that 8(z; w) 2 K2 ; jz − wj < δ ) jf(z) − f(w)j < 8f 2 F Observe that if there exists M > 0 such that 8 z 2 Ω ; 8 f 2 F, jf 0(z)j ≤ M, then F is equicontinuous. 1.2 Equivalence of the definitions Theorem (Montel's theorem): Suppose F is a locally uniformly bounded family of analytic functions in an open connected set Ω. Then F is normal. Proof : The proof of Montel's theorem can be decomposed into the following two steps: (1) Lemma: Suppose F is a family of analytic functions uniformly bounded on compact subsets of Ω. Then F is equicontinuous on every compact subset of Ω. To show this, let us consider a compact subset K of Ω. There exists r > 0 such that the set K~ defined by K~ := fz 2 C : d(z; K) ≤ 2rg is contained in Ω. Let M = sup jf(z)j f2F;z2K~ and (z; w) 2 K2 such that jz − wj < r. The closed disk D2r(z) is contained in Ω. If we call γ the boundary of this disk, we may write 1 Z f(ζ) f(ζ) 1 Z f(ζ)(z − w) f(z) − f(w) = − dζ = dζ 2πi γ ζ − z ζ − w 2πi γ (ζ − w)(ζ − z) 8 > 0, let δ = min (r=M; r). 1 4πr jz − wj < δ ) jf(z) − f(w)j ≤ Mjz − wj ≤ 2π 2r2 Hence F is equicontinuous on K. (2) Now that we know that F in Montel's theorem is equicontinuous on every compact subset of Ω, we can prove the theorem. 1 Let (fn)n2N be a sequence from F. Consider a sequence (wj)j2N which is everywhere dense in Ω. The sequence (fn(w1))n2N is bounded, so we can extract a convergent subsequence, and write the correspond- ing functions (fn;1)n2N.(fn;1(w2))n2N is bounded, so we can extract another convergent subsequence ∗ (fn;2)n2N. Repeating the process, we obtain the nested sequence (fn;k)k2N ;n2N such that (fn;k(wk))n2N converges for every wj with j 2 〚1; k〛. Let gn = fn;n be the diagonal sequence of functions. gn converges at all the points (wn)n2N. We now show that (gn)n 2 N converges uniformly on every compact subset K of Ω. Let K be such a compact subset. Since F is equicontinuous on K, 8 > 0 9δ > 0 such that 8(z; w) 2 K2 ; jz − wj < δ ) jf(z) − f(w)j < 3 N Since K is compact, we can select N points from (wj)j2N such that K ⊂ [p=1Dδ(wp). By construction of the functions gn, for that choice of , there exists N 2 N such that m; n ≥ N ) 8p 2 〚1;N〛; jgn(wp) − gm(wp)j < 3 Now, 8w 2 K, 9p0 such that w 2 Dδ(wp0 ). Then, if m; n ≥ N, jg (w) − g (w)j ≤ jg (w) − g (w )j + jg (w ) − g (w )j + jg (w ) − g (w)j ≤ + + = n m n n p0 n p0 m p0 m p0 m 3 3 3 from which we conclude that gn converges uniformly on K, as it is uniformly Cauchy on K. 2 The Riemann mapping theorem 2.1 Statement of the theorem Theorem (Riemann mapping theorem): Given any simply connected open set Ω which is not the whole plane, and a point z0 in Ω, there exists a unique analytic function f in Ω, normalized by the conditions 0 f(z0) = 0, f (z0) > 0, such that f defines a one-to-one mapping of Ω onto the disk jwj < 1. 2.2 Proving the theorem −1 Uniqueness: Suppose there are two such functions f1 and f2. Then f1 ◦ f2 is a one-to-one mapping of the −1 unit disk to itself. You will prove in Homework 9, using the lemma of Schwarz, that this implies that f1 ◦ f2 is a M¨obiustransformation of the form z − a S(z) = λ 1 − az with jλj = 1 and a 2 C such that jaj < 1. Then the normalization conditions S(0) = 0 ;S0(0) > 0 imply that S(z) = z, so that f1 ≡ f2. Existence: • Lemma: If Ω is a simply connected open set which is not C, there exists a one-to-one conformal map h :Ω 7! h(Ω) such that h(Ω) does not intersect a disk Dδ(w0) for some w0 2 C and δ > 0. To prove the lemma, observe that by hypothesis, there exists a 2 C n Ω. Hence '(z) = z − a is a nonvanishing analytic function on Ω. From Lecture 9, we then know that we can define an analytic function h on Ω such that h2(z) := z − a. 2 Observe that h does not take the same value twice, nor opposite values. Indeed if (z1; z2) 2 Ω are such that h(z1) = ±h(z2), then z1 − a = z2 − a ) z1 = z2. Now, by the open mapping theorem, 8z0 2 Ω, h(Ω) contains a disk Dδ(h(z0)). This means that h(Ω) does not contain the disk Dδ(−h(z0)), which proves the lemma, with w0 = −h(z0). • We now show that there are one-to-one, analytic functions g from Ω to D1(0) such that g(z0) = 0 and 0 g (z0) > 0. 2 To see that there is at least one such function, consider the function G0 on Ω defined by 0 δ jh (z0)j h(z0) h(z) − h(z0) G0(z) = 2 0 4 jh(z0)j h (z0) h(z) + h(z0) with δ and h as defined in the previous lemma. G0 is one-to-one since h is and G0 is obtained from h by a linear fractional transformation. 0 0 δ jh (z0)j G0(z0) = 0 ;G0(z0) = 2 > 0 8 jh(z0)j Finally, observe that δ 1 h(z) − h(z0) δ 1 2 8z 2 Ω ; jG0(z)j = = − < 1 4 jh(z0)j h(z) + h(z0) 4 h(z0) h(z) + h(z0) This proves our point. • The last step of the proof is to show that within the family F of functions g discussed previously, there exists an f with maximal derivative, and that this f has all the properties required for the mapping theorem. The key is to show that f is surjective. Observe that by Cauchy's estimate on Dr(z0) ⊂ Ω, 8 g 2 F, 1 jg0(z )j ≤ 0 r 0 0 The set fjg (z0)j ; g 2 Fg is therefore bounded, and has a supremum B = supg2F jg (z0)j. There exists 0 a sequence (gn)n2N 2 F such that gn(z0) ! B as n ! +1. Now, since jgnj < 1 on Ω, by Montel's theorem we know that there exists a subsequence (gnk ) of gn converging to a function f uniformly on every compact subset of Ω. By Weierstrass' theorem, f is analytic. Furthermore, f(z ) = lim g (z ) = 0 ; jf 0(z )j = lim jg0 (z )j = B > 0 0 nk 0 0 nk 0 nk!1 nk!1 From the latter result, we can say that f is not a constant function. Now, we take some z1 in Ω, and defineg ~nk := gnk (z) − gnk (z1).g ~nk is nonzero on Ω n fz1g. Hence, by ~ Hurwitz' theorem, f := f(z) − f(z1) is nonzero on Ω n fz1g. We have just proved that f is one-to-one on Ω. The remaining question is: is f onto? Suppose it is not: 9w0 2 D1(0) such that w0 2= f(Ω). Then, as before, we can consider the single-valued function F on Ω such that f(z) − w F 2(z) := 0 (1) 1 − w0f(z) F is one-to-one and satisfies jF (z)j < 1 for z 2 Ω. F can be normalized as follows: 0 jF (z0)j F (z) − F (z0) G(z) := 0 F (z0) 1 − F (z0)F (z) G is one-to-one, satisfies jG(z)j < 1 8z 2 Ω, and G(z0) = 0. Furthermore, jF 0(z )j jf 0(z )j 1 1 + jw j G0(z ) = 0 = 0 (1 − jw j2) = jf 0(z )j 0 > jf 0(z )j 0 2 p 0 0 p 0 1 − jF (z0)j 2 jw0j 1 − jw0j 2 jw0j This is a contradiction, so we may say that f is onto, which concludes the proof of the theorem 3 2.3 Returning to our definition of simple connectedness You may recall that in Lecture 9, we introduced an unusual definition of simple connectedness, only valid in R2, but very convenient for our purposes at the time.
Load more

Recommended publications
  • Projective Coordinates and Compactification in Elliptic, Parabolic and Hyperbolic 2-D Geometry
    PROJECTIVE COORDINATES AND COMPACTIFICATION IN ELLIPTIC, PARABOLIC AND HYPERBOLIC 2-D GEOMETRY Debapriya Biswas Department of Mathematics, Indian Institute of Technology-Kharagpur, Kharagpur-721302, India. e-mail: d [email protected] Abstract. A result that the upper half plane is not preserved in the hyperbolic case, has implications in physics, geometry and analysis. We discuss in details the introduction of projective coordinates for the EPH cases. We also introduce appropriate compactification for all the three EPH cases, which results in a sphere in the elliptic case, a cylinder in the parabolic case and a crosscap in the hyperbolic case. Key words. EPH Cases, Projective Coordinates, Compactification, M¨obius transformation, Clifford algebra, Lie group. 2010(AMS) Mathematics Subject Classification: Primary 30G35, Secondary 22E46. 1. Introduction Geometry is an essential branch of Mathematics. It deals with the proper- ties of figures in a plane or in space [7]. Perhaps, the most influencial book of all time, is ‘Euclids Elements’, written in around 3000 BC [14]. Since Euclid, geometry usually meant the geometry of Euclidean space of two-dimensions (2-D, Plane geometry) and three-dimensions (3-D, Solid geometry). A close scrutiny of the basis for the traditional Euclidean geometry, had revealed the independence of the parallel axiom from the others and consequently, non- Euclidean geometry was born and in projective geometry, new “points” (i.e., points at infinity and points with complex number coordinates) were intro- duced [1, 3, 9, 26]. In the eighteenth century, under the influence of Steiner, Von Staudt, Chasles and others, projective geometry became one of the chief subjects of mathematical research [7].
    [Show full text]
  • Hyperbolic Geometry
    Hyperbolic Geometry David Gu Yau Mathematics Science Center Tsinghua University Computer Science Department Stony Brook University [email protected] September 12, 2020 David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 1 / 65 Uniformization Figure: Closed surface uniformization. David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 2 / 65 Hyperbolic Structure Fundamental Group Suppose (S; g) is a closed high genus surface g > 1. The fundamental group is π1(S; q), represented as −1 −1 −1 −1 π1(S; q) = a1; b1; a2; b2; ; ag ; bg a1b1a b ag bg a b : h ··· j 1 1 ··· g g i Universal Covering Space universal covering space of S is S~, the projection map is p : S~ S.A ! deck transformation is an automorphism of S~, ' : S~ S~, p ' = '. All the deck transformations form the Deck transformation! group◦ DeckS~. ' Deck(S~), choose a pointq ~ S~, andγ ~ S~ connectsq ~ and '(~q). The 2 2 ⊂ projection γ = p(~γ) is a loop on S, then we obtain an isomorphism: Deck(S~) π1(S; q);' [γ] ! 7! David Gu (Stony Brook University) Computational Conformal Geometry September 12, 2020 3 / 65 Hyperbolic Structure Uniformization The uniformization metric is ¯g = e2ug, such that the K¯ 1 everywhere. ≡ − 2 Then (S~; ¯g) can be isometrically embedded on the hyperbolic plane H . The On the hyperbolic plane, all the Deck transformations are isometric transformations, Deck(S~) becomes the so-called Fuchsian group, −1 −1 −1 −1 Fuchs(S) = α1; β1; α2; β2; ; αg ; βg α1β1α β αg βg α β : h ··· j 1 1 ··· g g i The Fuchsian group generators are global conformal invariants, and form the coordinates in Teichm¨ullerspace.
    [Show full text]
  • The Riemann Mapping Theorem Christopher J. Bishop
    The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6.
    [Show full text]
  • Elliot Glazer Topological Manifolds
    Manifolds and complex structure Elliot Glazer Topological Manifolds • An n-dimensional manifold M is a space that is locally Euclidean, i.e. near any point p of M, we can define points of M by n real coordinates • Curves are 1-dimensional manifolds (1-manifolds) • Surfaces are 2-manifolds • A chart (U, f) is an open subset U of M and a homeomorphism f from U to some open subset of R • An atlas is a set of charts that cover M Some important 2-manifolds A very important 2-manifold Smooth manifolds • We want manifolds to have more than just topological structure • There should be compatibility between intersecting charts • For intersecting charts (U, f), (V, g), we define a natural transition map from one set of coordinates to the other • A manifold is C^k if it has an atlas consisting of charts with C^k transition maps • Of particular importance are smooth maps • Whitney embedding theorem: smooth m-manifolds can be smoothly embedded into R^{2m} Complex manifolds • An n-dimensional complex manifold is a space such that, near any point p, each point can be defined by n complex coordinates, and the transition maps are holomorphic (complex structure) • A manifold with n complex dimensions is also a real manifold with 2n dimensions, but with a complex structure • Complex 1-manifolds are called Riemann surfaces (surfaces because they have 2 real dimensions) • Holoorphi futios are rigid sie they are defied y accumulating sets • Whitney embedding theorem does not extend to complex manifolds Classifying Riemann surfaces • 3 basic Riemann surfaces • The complex plane • The unit disk (distinct from plane by Liouville theorem) • Riemann sphere (aka the extended complex plane), uses two charts • Uniformization theorem: simply connected Riemann surfaces are conformally equivalent to one of these three .
    [Show full text]
  • The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
    (June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication.
    [Show full text]
  • The Chazy XII Equation and Schwarz Triangle Functions
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 095, 24 pages The Chazy XII Equation and Schwarz Triangle Functions Oksana BIHUN and Sarbarish CHAKRAVARTY Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA E-mail: [email protected], [email protected] Received June 21, 2017, in final form December 12, 2017; Published online December 25, 2017 https://doi.org/10.3842/SIGMA.2017.095 Abstract. Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120{348] showed that the Chazy XII equation y000 − 2yy00 + 3y02 = K(6y0 − y2)2, K 2 C, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of K, can be expressed as y = a1w1 +a2w2 +a3w3 where wi solve the generalized Darboux{Halphen system. This relationship holds only for certain values of the coefficients (a1; a2; a3) and the Darboux{Halphen parameters (α; β; γ), which are enumerated in Table2. Consequently, the Chazy XII solution y(z) is parametrized by a particular class of Schwarz triangle functions S(α; β; γ; z) which are used to represent the solutions wi of the Darboux{Halphen system. The paper only considers the case where α + β +γ < 1. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple (P;^ Q;^ R^).
    [Show full text]
  • Chapter 2 Complex Analysis
    Chapter 2 Complex Analysis In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. We will extend the notions of derivatives and integrals, familiar from calculus, to the case of complex functions of a complex variable. In so doing we will come across analytic functions, which form the centerpiece of this part of the course. In fact, to a large extent complex analysis is the study of analytic functions. After a brief review of complex numbers as points in the complex plane, we will ¯rst discuss analyticity and give plenty of examples of analytic functions. We will then discuss complex integration, culminating with the generalised Cauchy Integral Formula, and some of its applications. We then go on to discuss the power series representations of analytic functions and the residue calculus, which will allow us to compute many real integrals and in¯nite sums very easily via complex integration. 2.1 Analytic functions In this section we will study complex functions of a complex variable. We will see that di®erentiability of such a function is a non-trivial property, giving rise to the concept of an analytic function. We will then study many examples of analytic functions. In fact, the construction of analytic functions will form a basic leitmotif for this part of the course. 2.1.1 The complex plane We already discussed complex numbers briefly in Section 1.3.5.
    [Show full text]
  • Math 372: Fall 2015: Solutions to Homework
    Math 372: Fall 2015: Solutions to Homework Steven Miller December 7, 2015 Abstract Below are detailed solutions to the homeworkproblemsfrom Math 372 Complex Analysis (Williams College, Fall 2015, Professor Steven J. Miller, [email protected]). The course homepage is http://www.williams.edu/Mathematics/sjmiller/public_html/372Fa15 and the textbook is Complex Analysis by Stein and Shakarchi (ISBN13: 978-0-691-11385-2). Note to students: it’s nice to include the statement of the problems, but I leave that up to you. I am only skimming the solutions. I will occasionally add some comments or mention alternate solutions. If you find an error in these notes, let me know for extra credit. Contents 1 Math 372: Homework #1: Yuzhong (Jeff) Meng and Liyang Zhang (2010) 3 1.1 Problems for HW#1: Due September 21, 2015 . ................ 3 1.2 SolutionsforHW#1: ............................... ........... 3 2 Math 372: Homework #2: Solutions by Nick Arnosti and Thomas Crawford (2010) 8 3 Math 372: Homework #3: Carlos Dominguez, Carson Eisenach, David Gold 12 4 Math 372: Homework #4: Due Friday, October 12, 2015: Pham, Jensen, Kolo˘glu 16 4.1 Chapter3,Exercise1 .............................. ............ 16 4.2 Chapter3,Exercise2 .............................. ............ 17 4.3 Chapter3,Exercise5 .............................. ............ 19 4.4 Chapter3Exercise15d ..... ..... ...... ..... ...... .. ............ 22 4.5 Chapter3Exercise17a ..... ..... ...... ..... ...... .. ............ 22 4.6 AdditionalProblem1 .............................. ............ 22 5 Math 372: Homework #5: Due Monday October 26: Pegado, Vu 24 6 Math 372: Homework #6: Kung, Lin, Waters 34 7 Math 372: Homework #7: Due Monday, November 9: Thompson, Schrock, Tosteson 42 7.1 Problems. ....................................... ......... 46 1 8 Math 372: Homework #8: Thompson, Schrock, Tosteson 47 8.1 Problems.
    [Show full text]
  • Math 311: Complex Analysis — Automorphism Groups Lecture
    MATH 311: COMPLEX ANALYSIS | AUTOMORPHISM GROUPS LECTURE 1. Introduction Rather than study individual examples of conformal mappings one at a time, we now want to study families of conformal mappings. Ensembles of conformal mappings naturally carry group structures. 2. Automorphisms of the Plane The automorphism group of the complex plane is Aut(C) = fanalytic bijections f : C −! Cg: Any automorphism of the plane must be conformal, for if f 0(z) = 0 for some z then f takes the value f(z) with multiplicity n > 1, and so by the Local Mapping Theorem it is n-to-1 near z, impossible since f is an automorphism. By a problem on the midterm, we know the form of such automorphisms: they are f(z) = az + b; a; b 2 C; a 6= 0: This description of such functions one at a time loses track of the group structure. If f(z) = az + b and g(z) = a0z + b0 then (f ◦ g)(z) = aa0z + (ab0 + b); f −1(z) = a−1z − a−1b: But these formulas are not very illuminating. For a better picture of the automor- phism group, represent each automorphism by a 2-by-2 complex matrix, a b (1) f(z) = ax + b ! : 0 1 Then the matrix calculations a b a0 b0 aa0 ab0 + b = ; 0 1 0 1 0 1 −1 a b a−1 −a−1b = 0 1 0 1 naturally encode the formulas for composing and inverting automorphisms of the plane. With this in mind, define the parabolic group of 2-by-2 complex matrices, a b P = : a; b 2 ; a 6= 0 : 0 1 C Then the correspondence (1) is a natural group isomorphism, Aut(C) =∼ P: 1 2 MATH 311: COMPLEX ANALYSIS | AUTOMORPHISM GROUPS LECTURE Two subgroups of the parabolic subgroup are its Levi component a 0 M = : a 2 ; a 6= 0 ; 0 1 C describing the dilations f(z) = ax, and its unipotent radical 1 b N = : b 2 ; 0 1 C describing the translations f(z) = z + b.
    [Show full text]
  • Riemann Mapping Theorem (4/10 - 4/15)
    Math 752 Spring 2015 Riemann Mapping Theorem (4/10 - 4/15) Definition 1. A class F of continuous functions defined on an open set G is called a normal family if every sequence of elements in F contains a subsequence that converges uniformly on compact subsets of G. The limit function is not required to be in F. By the above theorems we know that if F ⊆ H(G), then the limit function is in H(G). We require one direction of Montel's theorem, which relates normal families to families that are uniformly bounded on compact sets. Theorem 1. Let G be an open, connected set. Suppose F ⊂ H(G) and F is uniformly bounded on each compact subset of G. Then F is a normal family. ◦ Proof. Let Kn ⊆ Kn+1 be a sequence of compact sets whose union is G. Construction for these: Kn is the intersection of B(0; n) with the comple- ment of the (open) set [a=2GB(a; 1=n). We want to apply the Arzela-Ascoli theorem, hence we need to prove that F is equicontinuous. I.e., for " > 0 and z 2 G we need to show that there is δ > 0 so that if jz − wj < δ, then jf(z) − f(w)j < " for all f 2 F. (We emphasize that δ is allowed to depend on z.) Since every z 2 G is an element of one of the Kn, it suffices to prove this statement for fixed n and z 2 Kn. By assumption F is uniformly bounded on each Kn.
    [Show full text]
  • Mathematical Analysis, Second Edition
    PREFACE A glance at the table of contents will reveal that this textbooktreats topics in analysis at the "Advanced Calculus" level. The aim has beento provide a develop- ment of the subject which is honest, rigorous, up to date, and, at thesame time, not too pedantic.The book provides a transition from elementary calculusto advanced courses in real and complex function theory, and it introducesthe reader to some of the abstract thinking that pervades modern analysis. The second edition differs from the first inmany respects. Point set topology is developed in the setting of general metricspaces as well as in Euclidean n-space, and two new chapters have been addedon Lebesgue integration. The material on line integrals, vector analysis, and surface integrals has beendeleted. The order of some chapters has been rearranged, many sections have been completely rewritten, and several new exercises have been added. The development of Lebesgue integration follows the Riesz-Nagyapproach which focuses directly on functions and their integrals and doesnot depend on measure theory.The treatment here is simplified, spread out, and somewhat rearranged for presentation at the undergraduate level. The first edition has been used in mathematicscourses at a variety of levels, from first-year undergraduate to first-year graduate, bothas a text and as supple- mentary reference.The second edition preserves this flexibility.For example, Chapters 1 through 5, 12, and 13 providea course in differential calculus of func- tions of one or more variables. Chapters 6 through 11, 14, and15 provide a course in integration theory. Many other combinationsare possible; individual instructors can choose topics to suit their needs by consulting the diagram on the nextpage, which displays the logical interdependence of the chapters.
    [Show full text]
  • Conjugacy Classification of Quaternionic M¨Obius Transformations
    CONJUGACY CLASSIFICATION OF QUATERNIONIC MOBIUS¨ TRANSFORMATIONS JOHN R. PARKER AND IAN SHORT Abstract. It is well known that the dynamics and conjugacy class of a complex M¨obius transformation can be determined from a simple rational function of the coef- ficients of the transformation. We study the group of quaternionic M¨obius transforma- tions and identify simple rational functions of the coefficients of the transformations that determine dynamics and conjugacy. 1. Introduction The motivation behind this paper is the classical theory of complex M¨obius trans- formations. A complex M¨obiustransformation f is a conformal map of the extended complex plane of the form f(z) = (az + b)(cz + d)−1, where a, b, c and d are complex numbers such that the quantity σ = ad − bc is not zero (we sometimes write σ = σf in order to avoid ambiguity). There is a homomorphism from SL(2, C) to the group a b of complex M¨obiustransformations which takes the matrix ( c d ) to the map f. This homomorphism is surjective because the coefficients of f can always be scaled so that σ = 1. The collection of all complex M¨obius transformations for which σ takes the value 1 forms a group which can be identified with PSL(2, C). A member f of PSL(2, C) is simple if it is conjugate in PSL(2, C) to an element of PSL(2, R). The map f is k–simple if it may be expressed as the composite of k simple transformations but no fewer. Let τ = a + d (and likewise we write τ = τf where necessary).
    [Show full text]