DOCSLIB.ORG

Some Applications of the Kohn{Rossi Extension Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Applications of the Kohn{Rossi Extension Theorems SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS GEORGE MARINESCU Abstract We prove extension results for meromorphic functions by combining the KohnRossi extension theorems with Andreottis theory on the algebraic and analytic dep endence of meromorphic functions on pseudo concave manifolds Versions of KohnRossi theorems for pseudo convex domains are included Introduction A naive idea of extending a meromorphic function b eyond its domain of denition is to write as a quotient of two holomorphic functions and then extend these functions At least for holomorphic functions dened near the b oundary of complex manifolds this idea works as in the pro of of the KohnRossi extension theorem see Theorem and the pro of of subsequent Theorem But if is dened in a neighb ourho o d U of the complement Y n M of a smo oth domain M whose b oundary Levi form has at least one p ositive eigenvalue in a compact complex manifold Y we cannot apply this metho d Indeed U is pseudo concave and any holomorphic function on U is lo cally constant However it is rather simple to see that a meromorphic function on any complex manifold can b e written as a quotient of two holomorphic sections of an appropriate holomorphic line bundle We can thus hop e to extend the line bundle and its sections The situation in the ab ove example b ecomes particulary favourable in this resp ect if we assume Y to b e Moishezon Then by Andreottis theory it can b e shown that can b e written as a quotient of two holomorphic sections in some m tensor p ower E of a line bundle E Y with maximum Ko dairaIitaka m dimension Therefore we need to extend but the sections of E and this is done using the theorems of KohnRossi The situation b eing global we have to imp ose some supplementary hyp otheses namely one of the following see Theorem H M is q complete q n H There exists one nonconstant holomorphic function in the neigh b ourho o d of bM Mathematics Subject Classication Primary D A Secondary F Key words and phrases Extension of CR sections meromorphic function pseudo con cave manifold Moishezon manifold Ko dairaIitaka dimension The author was supp orted by a NATORoyal So ciety Fellowship at the University of Edinburgh He thanks the Department of Mathematics Statistics for its hospitality GEORGE MARINESCU H The holomorphic line bundle E of maximal Ko dairaIitaka dimen sion is trivial in the neighb ourho o d of M Note that using the Thullentyp e theorem of Siu see also Ivashkovich x and Shiman ab out the extension of a meromorphic map from the Hartogs gure to a p olydisc one can prove that if is a meromorphic map M with values in a Kahler manifold then has a meromorphic dened in Y n continuation into bM Our result can b e viewed as global counterpart of the previous theorem As for the b est global result let us mention the following theorem of Ivashkovich every meromorphic map from a domain D in a Stein manifold into a compact Kahler manifold K extends to a meromorphic b map from the envelop e of holomorphy D of D into K The KohnRossi theorems are consequences of the global regularity of the Neumann problem for domains M as ab ove Since the global regularity holds also for some class of pseudo convex domains we prove in the last section versions of those theorems for this class of domains Aknow ledgements I wish to acknowledge my debt to Professor Guennadi Henkin who is at the origin of the ideas used in this article He deserves a large share of the credit for any originality the present pap er may p ossess Meromorphic continuation Let us give a brief account of the extension theory develop ed by Kohn and Rossi Let M b e a smo oth domain in a complex manifold A smo oth function on bM is said to b e CauchyRiemann CR if its dierential vanishes on the subbundle C T bM T M A smo oth section over bM of a holomorphic vector bundle is said to b e CR if it is represented in lo cal holomorphic frames by vectors of CR functions In b oth cases we write f Of b course the restriction of a holomorphic section to bM is CR The crucial lemma dep ends on the natural duality b etween the free b oundary conditions imp osed by and is the formal adjoint and is the hilb ertian adjoint of and the solution of the Neumann problem To simplify matters let us take E V F merely a holomorphic section of the holomorphic vector bundle F over a neighb ourho o d of bM We dene the Ho dge op erator pq npnq E F E F by d v ol where the exterior pro duct is combined with the canonical pairing F F C Recall that r and on r forms Recall also that if a form M vanishes on bM then clearly b elongs to Dom with C co ecients on integration by parts and on such forms Consider a smo oth extension of to M and assume that the Neumann problem is solvable in bidegree n n for F which is the case if the Levi form of bM has one p ositive eigenvalue nn nn M F E M F the Neumann op erator Denote by N E Dene N E M F SOME APPLICATIONS OF THE KOHNROSSI EXTENSION THEOREMS nn Consider now E M F which has compact supp ort in M since has Clearly Dom and If is only CR then of course hasnt compact supp ort but we can construct an extension such that j alternatively Lemma of shows that in fact bM is equivalent to Dom for any extension Thus b N N N so that the Ho dge decomp osition given by the solution of the Neumann N H where H is the N H reduces to N nn orthogonal pro jector on the harmonic space H M F the space of smo oth forms Dom such that and Assume that nn H M F Then if we denote N we have and Therefore which implies On nn M F In this extreme case the other hand we have Dom E the b oundary conditions imp osed by the preceding relation are Dirichlet conditions j so that on bM We easily infer that bM in a neighb ourho o d of bM since real hyp ersurfaces are uniqueness sets for holomorphic functions and sections in holomorphic vector bundles Conversely if admits a holomorphic extension then and H van ish so that condition is necessary and sucient for the holomorphic extension Integrating by parts Z bM nn for H M F we obtain Lemma Let M be a relatively compact domain with smooth boundary in a complex manifold Y such that the Levi form of the boundary bM has at least one positive eigenvalue everywhere Let F be a holomorphic line bund le over Y Then a given smooth CR section of F over bM has a M which is holomorphic in M if and only if smooth extension over Z nn for any H M F bM On grounds of nite dimensionality of the harmonic space the Lemma entails see also the pro of of Theorem Theorem KohnRossi Let M as in Lemma and with connected boundary Then every CR function on bM has a holomorphic extension to al l of M Note that if there exist a nonconstant CR functions connectedness of bM is a sup eruous hyp othesis Theorem KohnRossi Let Y be a complex manifold possesing at least one nonconstant global holomorphic function Let M as in Lemma GEORGE MARINESCU Then for every holomorphic line bund le E on Y there exists a holomorphic line bund le L suct that every smooth CR section of E L over the boundary bM extends by a smooth section of E L over M holomorphic in M Of course these three statements remain true for holomorphic sections de ned near bM thanks to the uniqueness theorem Relying on these results we shall prove the following Theorem Let Y be a Moishezon manifold Y of dimension n Assume that M is a smooth domain in Y subject to one of the fol lowing hypothesis H M is q complete q n H The Levi form of bM has one positive eigenvalue everywhere and there exists one nonconstant holomorphic function in the neighbour hood of bM H The Levi form of bM has one positive eigenvalue everywhere and there exists a holomorphic line bund le E on Y of maximal Kodaira Iitaka dimension which is trivial in the neighbourhood of M Then any meromorphic function dened in a pseudoconcave neighbourhood U of bM eg a connected neighbourhood of Y n M there exists a mero morphic function on M U which agrees with it on U Observe that in view of the theorem of Siu and Ivashkovich quoted in the M Intro duction we may assume that is dened only on U Y n Recall that a compact connected complex manifold is said to b e Moishezon if the transcendence degree of its meromorphic function eld equals its com plex dimension The KodairaIitaka dimension E of a holomorphic line bundle on a compact complex manifold is the supremum of the generic rank m of the canonical rational maps dened by the nonzero sections in H E for m if any and E otherwise By a wellknown result of Siegel we have E deg tr K Y n dim Y where K Y is the eld of meromorphic functions on Y Therefore if Y p ossess a line bundle with maximum Ko dairaIitaka dimension then it is automatically Moishe zon Conversely if Y is Moishezon then there exists a nonsingular pro jective e mo dication comp osition of blowups of nonsingular centres Y Y e e if we consider an ample divisor H on Y then the line bundle asso ciated to e H H has maximum Ko dairaIitaka dimension Let us intro duce now the convexity notions used A real smo oth function dened on a complex manifold X of dimension n is said to b e q convex if its complex hessian i has at least n q p ositive eigenvalues at every p oint
Load more

Recommended publications
  • Topic 7 Notes 7 Taylor and Laurent Series
    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
    [Show full text]
  • Complex Analysis Class 24: Wednesday April 2
    Complex Analysis Math 214 Spring 2014 Fowler 307 MWF 3:00pm - 3:55pm c 2014 Ron Buckmire http://faculty.oxy.edu/ron/math/312/14/ Class 24: Wednesday April 2 TITLE Classifying Singularities using Laurent Series CURRENT READING Zill & Shanahan, §6.2-6.3 HOMEWORK Zill & Shanahan, §6.2 3, 15, 20, 24 33*. §6.3 7, 8, 9, 10. SUMMARY We shall be introduced to Laurent Series and learn how to use them to classify different various kinds of singularities (locations where complex functions are no longer analytic). Classifying Singularities There are basically three types of singularities (points where f(z) is not analytic) in the complex plane. Isolated Singularity An isolated singularity of a function f(z) is a point z0 such that f(z) is analytic on the punctured disc 0 < |z − z0| <rbut is undefined at z = z0. We usually call isolated singularities poles. An example is z = i for the function z/(z − i). Removable Singularity A removable singularity is a point z0 where the function f(z0) appears to be undefined but if we assign f(z0) the value w0 with the knowledge that lim f(z)=w0 then we can say that we z→z0 have “removed” the singularity. An example would be the point z = 0 for f(z) = sin(z)/z. Branch Singularity A branch singularity is a point z0 through which all possible branch cuts of a multi-valued function can be drawn to produce a single-valued function. An example of such a point would be the point z = 0 for Log (z).
    [Show full text]
  • Residue Theorem
    Topic 8 Notes Jeremy Orloff 8 Residue Theorem 8.1 Poles and zeros f z z We remind you of the following terminology: Suppose . / is analytic at 0 and f z a z z n a z z n+1 ; . / = n. * 0/ + n+1. * 0/ + § a ≠ f n z n z with n 0. Then we say has a zero of order at 0. If = 1 we say 0 is a simple zero. f z Suppose has an isolated singularity at 0 and Laurent series b b b n n*1 1 f .z/ = + + § + + a + a .z * z / + § z z n z z n*1 z z 0 1 0 . * 0/ . * 0/ * 0 < z z < R b ≠ f n z which converges on 0 * 0 and with n 0. Then we say has a pole of order at 0. n z If = 1 we say 0 is a simple pole. There are several examples in the Topic 7 notes. Here is one more Example 8.1. z + 1 f .z/ = z3.z2 + 1/ has isolated singularities at z = 0; ,i and a zero at z = *1. We will show that z = 0 is a pole of order 3, z = ,i are poles of order 1 and z = *1 is a zero of order 1. The style of argument is the same in each case. At z = 0: 1 z + 1 f .z/ = ⋅ : z3 z2 + 1 Call the second factor g.z/. Since g.z/ is analytic at z = 0 and g.0/ = 1, it has a Taylor series z + 1 g.z/ = = 1 + a z + a z2 + § z2 + 1 1 2 Therefore 1 a a f .z/ = + 1 +2 + § : z3 z2 z This shows z = 0 is a pole of order 3.
    [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]
  • Compex Analysis
    ( ! ) 1. The amplitu.e of a comple2 number isisis a) 0 b) 89: c) 8 .) :8 2.The amplitu.e of the quotient of two comple2 numbers is a) the sum of their amplitu.es b) the .ifference of their amplitu.es c) the pro.uct of their amplitu.es .) the quotient of their amplitu.es 3. The ? = A B then ?C = : : : : : : a) AB b) A B c) DB .) imaginary J ? K ?: J 4. If ? I ?: are any two comple2 numbers, then J ?.?: J isisis J ? K ?: J J ? K ?: J J ? K ?: J J J J J a)a)a) J ?: J b) J ?. J c) J ?: JJ? J .) J ? J J ?: J : : 555.5...MNO?P:Q: A B R = a) S b) DS c) T .) U 666.The6.The equation J? A UJ A J? D UJ = W represents a) Circle b) ellipse c) parparabolaabola .) hyperbola 777.The7.The equation J? A ZJ = J? D ZJ represents a) Circle b) ellipse c) parparabolaabola .) Real a2is 888.The8.The equation J? A J = ]:J? D J represents a) Circle b) ellipse c) parparabolaabola .) hyperbola 999.The9.The equation J?A:J A J?D:J = U represents a) Circle b) a st. line c) parabola .) hyperbohyperbolalalala 101010.The10.The equation J:? D J = J?D:J represents a) Circle b) a st. line c) parabola .) hyperbola : : 111111.The11.The equation J? D J A J?D:J = U represents a) Circle b) a st. line c) parabola .) hyperbola ?a 121212.The12.The equation !_ `?a b = c represents a) Circle b) a st.
    [Show full text]
  • COMPLEX ANALYSIS Notes Lent 2006 T. K. Carne
    Department of Pure Mathematics and Mathematical Statistics University of Cambridge COMPLEX ANALYSIS Notes Lent 2006 T. K. Carne. [email protected] c Copyright. Not for distribution outside Cambridge University. CONTENTS 1. ANALYTIC FUNCTIONS 1 Domains 1 Analytic Functions 1 Cauchy – Riemann Equations 1 2. POWER SERIES 3 Proposition 2.1 Radius of convergence 3 Proposition 2.2 Power series are differentiable. 3 Corollary 2.3 Power series are infinitely differentiable 4 The Exponential Function 4 Proposition 2.4 Products of exponentials 5 Corollary 2.5 Properties of the exponential 5 Logarithms 6 Branches of the logarithm 6 Logarithmic singularity 6 Powers 7 Branches of powers 7 Branch point 7 Conformal Maps 8 3. INTEGRATION ALONG CURVES 9 Definition of curves 9 Integral along a curve 9 Integral with respect to arc length 9 Proposition 3.1 10 Proposition 3.2 Fundamental Theorem of Calculus 11 Closed curves 11 Winding Numbers 11 Definition of the winding number 11 Lemma 3.3 12 Proposition 3.4 Winding numbers under perturbation 13 Proposition 3.5 Winding number constant on each component 13 Homotopy 13 Definition of homotopy 13 Definition of simply-connected 14 Definition of star domains 14 Proposition 3.6 Winding number and homotopy 14 Chains and Cycles 14 4 CAUCHY’S THEOREM 15 Proposition 4.1 Cauchy’s theorem for triangles 15 Theorem 4.2 Cauchy’s theorem for a star domain 16 Proposition 4.10 Cauchy’s theorem for triangles 17 Theorem 4.20 Cauchy’s theorem for a star domain 18 Theorem 4.3 Cauchy’s Representation Formula 18 Theorem 4.4 Liouville’s theorem 19 Corollary 4.5 The Fundamental Theorem of Algebra 19 Homotopy form of Cauchy’s Theorem.
    [Show full text]
  • MA 528 PRACTICE PROBLEMS 1. Find the Directional Derivative of F(X, Y)
    MA 528 PRACTICE PROBLEMS 1. Find the directional derivative of f(x, y)=5− 4x2 − 3y at (x, y) towards the origin −8x2 − 3y −8x − 3 8x2 +3y a. −8x − 3b.p c. √ d. 8x2 +3y e. p . x2 + y2 64x2 +9 x2 + y2 2. Find a vector pointing in the direction in which f(x, y, z)=3xy − 9xz2 + y increases most rapidly at the point (1, 1, 0). a. 3i +4j b. i + j c. 4i − 3j d. 2i + k e. −i + j. 1 3. Find a vector that is normal to the graph of the equation 2 cos(πxy) = 1 at the point ( 6 , 2). √ a. 6i + j b. − 3 i − j c. 12i + j d. j e. 12i − j. 4. Find an equation of the tangent plane to the surface x2 +2y2 +3z2 = 6 at the point (1, 1, −1). a. −x +2y +3z =2 b. 2x +4y − 6z =6 c. x − 2y +3z = −4 d. 2x +4y − 6z =6 e. x +2y − 3z =6. 5. Find an equation of the plane tangent to the graph of z = π + sin(πx2 +2y)when(x, y)=(2,π). a. 4πx +2y − z =9π b. 4x +2πy − z =10π c. 4πx +2πy + z =10π d. 4x +2πy − z =9π e. 4πx +2y + z =9π. 6. Are the following statements true or false? R 1. The line integral (x3 +2xy)dx +(x2 − y2)dy is independent of path in the xy-plane. R C 3 2 − 2 2. C (x +2xy)dx +(x y )dy = 0 for every closed oriented curve C in the xy-plane.
    [Show full text]
  • 20. Isolated Singularities Definition 20.1. Let F : U −→ C Be A
    20. Isolated Singularities Definition 20.1. Let f : U −! C be a holomorphic function on a region U. Let a2 = U. We say that a is an isolated singularity of f if U contains a punctured neighbourhood of a. Note that Log z does not have an isolated singularity at 0, since we have to remove all of (−∞; 0] to get a continuous function. By contrast its derivative 1=z is holomorphic except at 0 and so it has an isolated singularity at 0. Suppose that f has an isolated singularity at a. As a punctured neighbourhood of a is a special type of annulus, f has a Laurent ex- pansion centred at a, 1 X k f(z) = ak(z − a) ; k=−∞ valid for 0 < jz − aj < r; for some real r. The behaviour at a is dictated by the negative part of the Laurent expansion. Definition 20.2. If f has an isolated singularity at a and all of the coefficients ak of the Laurent expansion 1 X k f(z) = ak(z − a) ; k=−∞ vanish if k < 0, then we say that f has a removable singularity. If f has a removable singularity then in fact we can extend f to a holomorphic function in a neighbourhood of a. Indeed, the Laurent ex- pansion of f is a power series expansion, and this defines a holomorphic function in a neighbourhood of a. Example 20.3. The function sin z z has a removable singularity at a. 1 Indeed, sin z 1 z3 z3 = z − + + ::: z z 3! 5! z2 z4 = 1 − + + :::; 3! 5! is the Laurent series expansion of sin z=z.
    [Show full text]
  • Complex Analysis Fall 2007 Homework 11: Solutions
    Complex Analysis Fall 2007 Homework 11: Solutions k 3.3.16 If f has a zero of multiplicity k at z0 then we can write f(z) = (z − z0) φ(z), where 0 k−1 φ is analytic and φ(z0) 6= 0. Differentiating this expression yields f (z) = k(z − z0) φ(z) + k 0 (z − z0) φ (z). Therefore 0 k−1 k 0 0 f (z) k(z − z0) φ(z) + (z − z0) φ (z) k φ (z) = k = + . f(z) (z − z0) φ(z) z − z0 φ(z) 0 Since φ(z0) 6= 0, the function φ (z)/φ(z) is analytic at z0 and therefore has a convergent Taylor series in a neighborhood of z0. Thus ∞ f 0(z) k φ0(z) k X = + = + a (z − z )n. f(z) z − z φ(z) z − z n 0 0 0 n=0 for all z in a deleted neighborhood of z0. Uniqueness of such expressions implies that this is 0 0 the Laurent series for f /f in a deleted neighborhood of z0 and therefore f /f has a simple pole at z0 with residue k. 3.3.20(a) The closure of a set A is the intersection of all the closed sets that contain A. It is not hard to show that an element belongs to the closure of A if and only if every neighborhood of that point intersects A. Therefore, what we need to prove the following: given any w ∈ C and any > 0, there exists z ∈ U so that f(z) ∈ D(w; ).
    [Show full text]
  • A Concise Course in Complex Analysis and Riemann Surfaces Wilhelm Schlag
    A concise course in complex analysis and Riemann surfaces Wilhelm Schlag Contents Preface v Chapter 1. From i to z: the basics of complex analysis 1 1. The field of complex numbers 1 2. Differentiability and conformality 3 3. M¨obius transforms 7 4. Integration 12 5. Harmonic functions 19 6. The winding number 21 7. Problems 24 Chapter 2. From z to the Riemann mapping theorem: some finer points of basic complex analysis 27 1. The winding number version of Cauchy’s theorem 27 2. Isolated singularities and residues 29 3. Analytic continuation 33 4. Convergence and normal families 36 5. The Mittag-Leffler and Weierstrass theorems 37 6. The Riemann mapping theorem 41 7. Runge’s theorem 44 8. Problems 46 Chapter 3. Harmonic functions on D 51 1. The Poisson kernel 51 2. Hardy classes of harmonic functions 53 3. Almost everywhere convergence to the boundary data 55 4. Problems 58 Chapter 4. Riemann surfaces: definitions, examples, basic properties 63 1. The basic definitions 63 2. Examples 64 3. Functions on Riemann surfaces 67 4. Degree and genus 69 5. Riemann surfaces as quotients 70 6. Elliptic functions 73 7. Problems 77 Chapter 5. Analytic continuation, covering surfaces, and algebraic functions 79 1. Analytic continuation 79 2. The unramified Riemann surface of an analytic germ 83 iii iv CONTENTS 3. The ramified Riemann surface of an analytic germ 85 4. Algebraic germs and functions 88 5. Problems 100 Chapter 6. Differential forms on Riemann surfaces 103 1. Holomorphic and meromorphic differentials 103 2. Integrating differentials and residues 105 3.
    [Show full text]
  • Computational Complex Analysis Book
    COMPUTATIONAL COMPLEX ANALYSIS VERSION 0.0 Frank Jones 2018 1 Desmos.com was used to generate graphs. © 2017 Frank Jones. All rights reserved. Table of Contents PREFACE ......................................................................................................................................................... i CHAPTER 1: INTRODUCTION ........................................................................................................................ 1 SECTION A: COMPLEX NUMBERS .............................................................................................................. 1 2 SECTION B: LINEAR FUNCTIONS ON ℝ ................................................................................................ 20 SECTION C: COMPLEX DESCRIPTION OF ELLIPSES ................................................................................... 23 CHAPTER 2: DIFFERENTIATION .................................................................................................................. 28 SECTION A: THE COMPLEX DERIVATIVE .................................................................................................. 28 SECTION B: THE CAUCHY-RIEMANN EQUATION ..................................................................................... 31 SECTION C: HOLOMORPHIC FUNCTIONS ................................................................................................ 38 SECTION D: CONFORMAL TRANSFORMATIONS ...................................................................................... 41 SECTION E: (COMPLEX)
    [Show full text]
  • Class 2/14 1 Singularities
    Math 752 Spring 2011 Class 2/14 1 Singularities Definition 1. f has an isolated singularity at z = a if there is a punctured disk B(a, R)\{a} such that f is defined and analytic on this set, but not on the full disk. a is called removable singularity if there is an analytic g : B(a, R) → C such that g(z) = f(z) for 0 < |z − a| < R. Investigation of removable singularities. Example 1. f(z) = z−1 sin z has a removable singularity at the origin. To see this, divide the power expansion of sin z by z. Theorem 1. If f has an isolated singularity at a, then the point z = a is a removable singularity if and only if lim(z − a)f(z) = 0. z→a Proof. Need to show ⇐=: Suppose that f is analytic for 0 < |z − a| < R. Define g(z) = (z − a)f(z) for z 6= a and g(a) = 0, and note that by assumption g is continuous at a. We will show that g is analytic in B(a, R). Since an analytic function has zeros of integer order, it then follows that f is analytic in this disk. We apply Morera’s theorem. Let T be a triangular path in B(a, R), R enclosing the triangle ∆. If a∈ / ∆, then T g = 0. If a is a vertex of ∆, partition ∆ into two triangles, one a very small one with vertex a, and use continuity of g on the smaller to bound the contribution to the integral by ε.
    [Show full text]