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Riemann Mapping Theorem Riemann Mapping Theorem Juan Kim 2020 Abstract The purpose of this report is to reproduce the proof of Riemann Mapping Theorem. We assume the readers are familiar with the basic results in real and complex analysis upto and including the Residue Theorem. Following the approach of Riesz and Fejer, we define F to be a family of holomorphic injective functions, uniformly bounded by 1 on a given domain. After showing that F is not empty by constructing a function in F, we will use Montel's theorem to show that there is a sequence in F that converges locally uniformly to some holomorphic injective function in f : U ! B1(0) where at z0 0 such that fn(z0) = 0 for all n, jfn(z0)j ≤ jf (z0)j. We then use Hurwitz's theorem to show injectivity of f and proof by contradiction to show that such f is surjective to the unit disk. Contents 1 Notations 2 2 Basic Results 3 2.1 General Cauchy Theorem . 3 2.2 Properties of Mobius Transformations . 4 2.3 Sequences of Functions . 4 3 Montel's Theorem 6 4 Hurwitz's Theorem 8 5 Proof of the Riemann Mapping Theorem 11 6 Bibliography 16 1 1 Notations Here we list some of notation that will be used throughout this report. If A is a set, A will denote the closure of A. The unit disk is denoted by B1(0) = fz 2 C : jzj < 1g. If f : U ! C is a function and A ⊂ U a set, then f(A) will denote image set of f. If γ :[a; b] ! C is a curve, γ∗ := γ([a; b]) is the image set of the curve. The sequence (an)n2N will be written (an). P1 P The series n=0 an will be denoted an. If (fn) is a sequence of functions then it's subsequence (fnk )k2N will be denoted (fnk ) or i sometimes (fn) to mean the ith subsequence of (fn). If [a; b] is a real interval and P = fa0:::ang a partition of [a; b], we denote the ith subinterval by ∆i := [ai−1; ai]: The annulus centred at z0 with inner radius r1, outer radius r2 is the set A(z0; r1; r2) = fz 2 C : r1 < jz − z0j < r2g. 2 2 Basic Results Here we state the relevant definitions and some of the basic results from real and complex analysis for the sake of completeness. These results will be used throughout the proofs of some of the main theorems involved in proving the Riemann Mapping Theorem. 2.1 General Cauchy Theorem Definition 1. Let U be an open subset of C and γ1 :[a0; b0] ! U and γ2 :[a1; b1] ! U be closed curves in U. We say γ1 is homotopic to γ2 if there exists a continuous function φ : [0; 1] × [0; 1] ! U such that φ(t; 0) = γ1(a0 + t(b0 − a0)) for all t 2 [0; 1] φ(t; 1) = γ1(a1 + t(b1 − a1)) for all t 2 [0; 1] φ(0; s) = φ(1; s) for all s 2 [0; 1]. The function φ is called a homotopy from γ1 to γ2 in U. If γ1 is homotopic to a constant curve, then γ1 is said to be null homotopic in U. Definition 2. a simply connected set is a set in which every closed, pieces wise smooth curve is null homotopic. Theorem 1 (Homotopy Theorem). Let U ⊂ C be an open set and f : U ! C a holomorphic function. If γ1 and γ2 are curves in U, homotopic to each other then Z Z f(z)dz = f(z)dz γ1 γ2 . The proof of the Homotopy Theorem is not included in this article but can be found in page 93 of, Complex Analysis by E. M. Stein and R. Sharkarchi. As a Corollary, we obtain the Cauchy's theorem for simply connected sets. Theorem 2 (Cauchy). Let S ⊂ C be an open and simply connected set, f : S ! C a holomorphic function and γ a closed piecewise smooth curve in U, then Z f(z)dz = 0 γ . Theorem 3 (Morera). Suppose f is a continuous complex function in an open set U such that Z f(z)dz = 0 @4 for every closed triangle 4 2 U. Then f is holomorphic 3 The proof of Morera's Theorem can be found in page 224 of Real and Complex Anal- ysis 2nd edition by W. Rudin. Definition 3. Let 0 ≤ r1 < r2 ≤ 1 and z0 2 C. Let f : A(z0; r1; r2) ! C be a P1 −k P1 k holomorphic function. Suppose 1 α−k(z − z0) + 0 αk(z − z0) is the Laurent Series of f about z0. Then α−1 is the residue of f at z0 denoted α−1. Theorem 4 (Residue Theorem). Let U ⊂ C be an open set and A ⊂ U. Suppose U=A is open. Let f : U=A ! C be a holomorphic function. Suppose f has a isolated singularity at each a 2 A. Let γ be a closed piecewise smooth curve in U=A which is null homotopic in U. Then N 1 Z X f(z)dz = (Res f)Ind (a ) 2πi ak γ k γ k=1 . 2.2 Properties of Mobius Transformations Mobius transformations are used to construct an example of a holomorphic injective function bounded by 1 in a given open simply connected set. It is necessary to show that the family of functions on U that are holomorphic, injective and bounded by 1 is not empty, for if such family was empty, the Riemann mapping theorem is trivially false. Definition 4. Let U ! C be an open, simply connected set. The function f : U ! C is az+b a mobius transformation if for some a; b; c; d 2 C, f(z) = cz+d . −d Proposition 1. Let U ! C be an open set and a; b; c; d 2 C such that c 62 U and az+b ad − bc 6= 0. If f : U ! C is defined by f(z) = cz+d for all z 2 C, then f is injective and holomorphic in U. Proof. Holomorphicity is trivial since the mapping z 7! az + b and z 7! cz + d are holo- −d morphic and for all z 2 C, cz + d 6= 0 as c 62 U. Suppose that for some w; z 2 C, f(z) = f(w), then we get (ad − bc)z = (ad − bc)w. By hypothesis (ad − bc) 6= 0 so z = w. So f is injective. 2.3 Sequences of Functions In defining a non-empty family of holomorphic, injective functions, bounded by 1, we wish to show that such family always contains a sequence of functions converging to a function with the same properties (such is the content of Montel's Theorem and Weier- strass Theorem). Here we give the sufficient conditions for the limit of a sequence to inherit the properties of the functions in that sequence. Definition 5. Let U ⊂ C be open, (fn)N a sequence of functions on U. The sequence (fn)N is said to be pointwise convergent to f if for each z 2 U, the sequence (fn(z))N converges to f(z). 4 Definition 6. Let U ⊂ C be open, and fn : U ! C for each n 2 N functions on U. The sequence (fn)N is said to be uniformly convergent to f in U if for each ", there exists N 2 N such that for each z 2 U, for all n 2 N with N < n we have jfn(z) − f(z)j < ". Definition 7. Let U ⊂ C be open, and fn : U ! C for each n 2 N functions on U. The sequence (fn)N is said to be locally uniformly convergent to f if for each z 2 U there exist 0 < r such that Br(z) ⊂ U and the sequence (fn)N converges uniformly to f in Br(z). Theorem 5. Let [a; b] be a closed interval in R and (fn)N a sequence of integrable functions from [a; b] to C. If (fn)N converges uniformly to f :[a; b] ! C, then f is R b R b integrable and a f(x)dx = limn!1 a fn(x)dx. Proof. Since the integral of complex valued function f of real variables are defined in- terms of the individual integral of the real and imaginary components of f, it is sufficient to prove the case for the sequence of real valued functions of real variable. Suppose (fn) is a sequence of integrable functions from the interval [a,b] to R, uni- formly convergent to f :[a; b] ! R. Let " be given and N 2 N such that for all x 2 [a; b], N < n, we have jfn(x) − f(x)j < ". Pick N < n and let P = fa0; :::; ang be a partition of [a; b] such that U(P; fn) − L(P; fn) < ". We have fn(x) − " < f(x) < fn(x) + " for all x 2 [a; b] and thus for each subinterval ∆i we have infx2∆i fn(x) − " < infx2∆i f(x) ≤ supx2∆i f(x) < supx2∆i fn(x) + ". R − R We Thus have 0 ≤ f(x)dx− − f(x)dx ≤ U(P; f)−L(P; f) < U(P; fn +")−L(P; fn − ") = U(P; fn) − L(P; fn) + "(b − a) < "(1 + b − a). R − R Since 0 < " was arbitrary, f(x)dx = − f(x)dx and so f is Rieman Integrable. For each n 2 N let "n = supx2[a;b]jf(x) − fn(x)j then limn!1"n = 0.
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