DOCSLIB.ORG

Chapter 1 Holomorphic and Real Analytic Calculus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

This version: 28/02/2014 Chapter 1 Holomorphic and real analytic calculus In this chapter we develop basic analysis in the holomorphic and real analytic settings. We do this, for the most part, simultaneously, and as a result some of the ways we do things are a little unconventional, especially when compared to the standard holomorphic treatments in, for example, texts like [Fritzsche and Grauert 2002, Gunning and Rossi 1965,H ormander¨ 1973, Krantz 1992, Laurent-Thiebaut´ 2011, Range 1986, Taylor 2002]. We assume, of course, a thorough acquaintance with real analysis such as one might find in [Abraham, Marsden, and Ratiu 1988, Chapter 2] and single variable complex analysis such as one might find in [Conway 1978]. 1.1 Holomorphic and real analytic functions Holomorphic and real analytic functions are defined as being locally prescribed by a convergent power series. We, therefore, begin by describing formal (i.e., not depending on any sort of convergence) power series. We then indicate how the usual notion of a Taylor series gives rise to a formal power series, and we prove Borel’s Theorem which says that, in the real case, all formal power series arise as Taylor series. This leads us to consider convergence of power series, and then finally to consider holomorphic and real analytic functions. Much of what we say here is a fleshing out of some material from Chapter 2 of [Krantz and Parks 2002], adapted to cover both the holomorphic and real analytic cases simultaneously. 1.1.1 Fn Because much of what we say in this chapter applies simultaneously to the real and complex case, we shall adopt the convention of using the symbol F when we wish to refer to one of R or C. We shall use x to mean the absolute value (when F = R) or j j the complex modulus (when F = C). In like manner, we shall denote by x¯ the complex conjugate of x if F = C and x¯ = x if F = R. We use Xn x; y = xj y¯ j h i j=1 2 1 Holomorphic and real analytic calculus 28/02/2014 to denote the standard inner product and n X 1=2 2 x = xj k k j j j=1 to denote the standard norm on Fn in both cases. We shall need notions of balls and disks in Fn. Balls are defined in the usual way. n Given x0 F and r R>0, the open ball with radius r and centre x0 is 2 2 n n B (r; x0) = x F x x0 < r ; f 2 j k − k g with n n B (r; x0) = x F x x0 r f 2 j k − k ≤ g n n similarly denoting the closed ball. For x0 F and for r R , denote 2 2 >0 n n D (r; x0) = x F xj x0j < rj; j 1;:::; n ; f 2 j j − j 2 f gg which we call the open polydisk of radius r and centre x0. The closure of the open polydisk, interestingly called the closed polydisk, is denoted by n n D (r; x0) = x F xj x0j rj; j 1;:::; n : f 2 j j − j ≤ 2 f gg n n n If r R>0 then we denote rˆ = (r;:::; r) R>0 so that D (rˆ; x0) and D (rˆ; x0) denote polydisks2 whose radii in all components are2 equal. If U Fn is open, if A U, and if f : U Fm is continuous, we denote ⊆ ⊆ ! f A = sup f(x) x A : (1.1) k k fk k j 2 g A domain in Fn is a nonempty connected open set. m n m r A map f : U F from an open subset U F to F will be of class C , r Z 0 , !r ⊆ 2 ≥ [f1g if it is of class C in the real variable sense, noting that Cn R2n. ' 1.1.2 Multi-index and partial derivative notation n A multi-index is an element of Z 0. For a multi-index I we shall write I = (i1;:::; in). We introduce the following notation:≥ 1. I = i + + in; j j 1 ··· 2. I! = i ! in!; 1 ··· I i1 in n 3. x = x1 xn for x = (x1;:::; xn) F . ··· 2 n n Note that the elements e1;:::; en of the standard basis for F are in Z 0, so we shall n ≥ think of these vectors as elements of Z 0 when it is convenient to do so. The following property of the set of≥ multi-indices will often be useful. 28/02/2014 1.1 Holomorphic and real analytic functions 3 1.1.1 Lemma (Cardinality of sets of multi-indices) For n Z>0 and m Z 0, 2 2 ≥ ! n n + m 1 card I Z 0 I = m = − : f 2 ≥ j j j g n 1 − Proof We begin with an elementary lemma. m ! ! X n + j 1 n + m 1 Sublemma For n Z>0 and m Z 0, − = . 2 2 ≥ n 1 n j=0 − Proof Recall that for j; k Z 0 with j k we have 2 ≥ ≤ ! k k! = : j j!(k j)! − We claim that if j; k Z>0 satisfy j k then 2 ≤ ! ! ! k k k + 1 + = : j j 1 j − This is a direct computation: k! k! (k j + 1)k! jk! + = − + j!(k j)! (j 1)!(k j + 1)! (k j + 1)j!(k j)! j(j 1)!(k j + 1)! − − − − − − − (k j + 1)k! + jk! = − j!(k j + 1)! − ! (k + 1)k! (k + 1)! k + 1 = = = : j!((k + 1) j)! j!((k + 1) j)! j − − Now we have m ! m ! m ! m ! X n + j 1 X n + j 1 X n + j X n + j 1 − = 1 + − = 1 + − n 1 n 1 n − n j=0 − j=1 − j=1 j=1 ! ! ! n + m n n + m = 1 + = ; n − n n as desired. H We now prove the lemma by induction on n. For n = 1 we have ! m card j Z 0 j = m = 1 = ; f 2 ≥ j g 0 which gives the conclusions of the lemma in this case. Now suppose that the lemma k+1 holds for n 1;:::; k . If I Z 0 satisfies I = m, then write I = (i1;:::; ik; ik+1) and take 2 f g 2 ≥ j j 4 1 Holomorphic and real analytic calculus 28/02/2014 I = (i ;:::; i ). If i = j 0; 1;:::; m then I = m j. Thus 0 1 k k+1 2 f g j 0j − Xm k+1 n card I Z 0 I = m = card I0 Z 0 I0 = m j f 2 ≥ j j j g f 2 ≥ j j j − g j=0 m ! m ! X k + m j + 1 X k + j 1 = − = 0 − k 1 k 1 j=0 − j0=0 − ! ! k + m (k + 1) + m 1 = = − ; k (k + 1) 1 − using the sublemma in the penultimate step. This proves the lemma by induction. Multi-index notation is also convenient for representing partial derivatives of multi- variable functions. Let us start from the ground up. Let (e1;:::; en) be the standard n n n basis for F and denote by (α1;:::; αn) the dual basis for (F )∗. Let U F and let ⊆ n f : U F. The F-derivative of f at x0 U is the unique F-linear map D f (x0) L(F ; F) for which! 2 2 f (x) f (x0) D f (x0)(x x0) lim − − − = 0; x x0 x x ! − 0 provided that such a linear map indeed exists. Higher F-derivatives are defined k k recursively. The kth F-derivative of f at x0 we denote by D f (x0), noting that D f (x0) k n 2 Lsym(F ; F) is a symmetric k-multilinear map [see Abraham, Marsden, and Ratiu 1988, Proposition 2.4.14]. If Dk f (x) exists for every x U and if the map x Dk f (x) is k 2 7! continuous, f is of F-class C or k-times continuously F-differentiable. This slightly awkward and nonstandard notation will be short-lived, and is a consequence of our trying to simultaneously develop the real and complex theories. k We denote by L (Fn; F) the set of k-multilinear maps with its usual basis (αj αj ) j ;:::; jk 1;:::; n : f 1 ⊗ · · · ⊗ k j 1 2 f gg Thinking of D f (x0) as a multilinear map, forgetting about its being symmetric, we write Xn k k @ f D f (x0) = (x0)αj1 αjk : @xj1 @xjk ⊗ · · · ⊗ j1;:::;jk=1 ··· Let us introduce another way of writing the F-derivative. For j1;:::; jk 1;:::; n , n 2 f g define I Z 0 by letting im Z 0 be the number of times m 1;:::; n appears in 2 ≥ 2 ≥ 2 f g the list of numbers j1;:::; jk. We recall from Section F.2.3 the product between k l A L (Fn; F) and B L (Fn; F) given by 2 sym 2 sym (k + l)! A B = Sym (A B); k!l! k+l ⊗ m where, for C L (Fn; F), 2 sym 1 X Sym (C)(v ;:::; v ) = C(v ;:::; v ): m 1 m m! σ(1) σ(m) σ Sm 2 28/02/2014 1.1 Holomorphic and real analytic functions 5 Then, as discussed in Section F.2.4, we can also write X 1 @k f Dk f x x i1 in ( 0) = i i ( 0)α1 αn i1! in! @x 1 @x n · · · i1;:::;in Z 0 1 n 2 ≥ ··· ··· i + +in=k 1 ··· n For I = (i1;:::; in) Z 0, we write 2 ≥ I I @j j f D f (x0) = (x0): @xi1 @xin 1 ··· n We may also write this in a different way: I I @j j f D f (x0) = (x0): @x1 @x1 @xn @xn | {z··· } ··· | {z··· } i1 times in times Indeed, because of symmetry of the F-derivative, for any collection of numbers j1;:::; j I 1;:::; n for which k occurs ik times for each k 1;:::; n , we have j j 2 f g 2 f g I I @j j f D f (x0) = (x0): @xj1 @xj I ··· j j In any case, we can also write X 1 Dk f (x ) = DI f (x )αi1 αin 0 I! 0 1 · · · n I Zn 2 0 I =≥k j j We shall freely interchange the various partial F-derivative notations discussed above, depending on what we are doing.
Recommended publications
  • Class 1/28 1 Zeros of an Analytic Function

    Class 1/28 1 Zeros of an Analytic Function

    Math 752 Spring 2011 Class 1/28 1 Zeros of an analytic function Towards the fundamental theorem of algebra and its statement for analytic functions. Definition 1. Let f : G → C be analytic and f(a) = 0. a is said to have multiplicity m ≥ 1 if there exists an analytic function g : G → C with g(a) 6= 0 so that f(z) = (z − a)mg(z). Definition 2. If f is analytic in C it is called entire. An entire function has a power series expansion with infinite radius of convergence. Theorem 1 (Liouville’s Theorem). If f is a bounded entire function then f is constant. 0 Proof. Assume |f(z)| ≤ M for all z ∈ C. Use Cauchy’s estimate for f to obtain that |f 0(z)| ≤ M/R for every R > 0 and hence equal to 0. Theorem 2 (Fundamental theorem of algebra). For every non-constant polynomial there exists a ∈ C with p(a) = 0. Proof. Two facts: If p has degree ≥ 1 then lim p(z) = ∞ z→∞ where the limit is taken along any path to ∞ in C∞. (Sometimes also written as |z| → ∞.) If p has no zero, its reciprocal is therefore entire and bounded. Invoke Liouville’s theorem. Corollary 1. If p is a polynomial with zeros aj (multiplicity kj) then p(z) = k k km c(z − a1) 1 (z − a2) 2 ...(z − am) . Proof. Induction, and the fact that p(z)/(z − a) is a polynomial of degree n − 1 if p(a) = 0. 1 The zero function is the only analytic function that has a zero of infinite order.
  • Informal Lecture Notes for Complex Analysis

    Informal Lecture Notes for Complex Analysis

    Informal lecture notes for complex analysis Robert Neel I'll assume you're familiar with the review of complex numbers and their algebra as contained in Appendix G of Stewart's book, so we'll pick up where that leaves off. 1 Elementary complex functions In one-variable real calculus, we have a collection of basic functions, like poly- nomials, rational functions, the exponential and log functions, and the trig functions, which we understand well and which serve as the building blocks for more general functions. The same is true in one complex variable; in fact, the real functions we just listed can be extended to complex functions. 1.1 Polynomials and rational functions We start with polynomials and rational functions. We know how to multiply and add complex numbers, and thus we understand polynomial functions. To be specific, a degree n polynomial, for some non-negative integer n, is a function of the form n n−1 f(z) = cnz + cn−1z + ··· + c1z + c0; 3 where the ci are complex numbers with cn 6= 0. For example, f(z) = 2z + (1 − i)z + 2i is a degree three (complex) polynomial. Polynomials are clearly defined on all of C. A rational function is the quotient of two polynomials, and it is defined everywhere where the denominator is non-zero. z2+1 Example: The function f(z) = z2−1 is a rational function. The denomina- tor will be zero precisely when z2 = 1. We know that every non-zero complex number has n distinct nth roots, and thus there will be two points at which the denominator is zero.
  • Lecture Notes in Mathematics

    Lecture Notes in Mathematics

    Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universit~it und Max-Planck-lnstitut for Mathematik, Bonn - voL 5 Adviser: E Hirzebruch 1111 Arbeitstagung Bonn 1984 Proceedings of the meeting held by the Max-Planck-lnstitut fur Mathematik, Bonn June 15-22, 1984 Edited by E Hirzebruch, J. Schwermer and S. Suter I IIII Springer-Verlag Berlin Heidelberg New York Tokyo Herausgeber Friedrich Hirzebruch Joachim Schwermer Silke Suter Max-Planck-lnstitut fLir Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3, Federal Republic of Germany AMS-Subject Classification (1980): 10D15, 10D21, 10F99, 12D30, 14H10, 14H40, 14K22, 17B65, 20G35, 22E47, 22E65, 32G15, 53C20, 57 N13, 58F19 ISBN 3-54045195-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15195-8 Springer-Verlag New York Heidelberg Berlin Tokyo CIP-Kurztitelaufnahme der Deutschen Bibliothek. Mathematische Arbeitstagung <25. 1984. Bonn>: Arbeitstagung Bonn: 1984; proceedings of the meeting, held in Bonn, June 15-22, 1984 / [25. Math. Arbeitstagung]. Ed. by E Hirzebruch ... - Berlin; Heidelberg; NewYork; Tokyo: Springer, 1985. (Lecture notes in mathematics; Vol. 1t11: Subseries: Mathematisches I nstitut der U niversit~it und Max-Planck-lnstitut for Mathematik Bonn; VoL 5) ISBN 3-540-t5195-8 (Berlin...) ISBN 0-387q5195-8 (NewYork ...) NE: Hirzebruch, Friedrich [Hrsg.]; Lecture notes in mathematics / Subseries: Mathematischee Institut der UniversitAt und Max-Planck-lnstitut fur Mathematik Bonn; HST This work ts subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
  • Topic 7 Notes 7 Taylor and Laurent Series

    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.
  • Bounded Holomorphic Functions on Finite Riemann Surfaces

    Bounded Holomorphic Functions on Finite Riemann Surfaces

    BOUNDED HOLOMORPHIC FUNCTIONS ON FINITE RIEMANN SURFACES BY E. L. STOUT(i) 1. Introduction. This paper is devoted to the study of some problems concerning bounded holomorphic functions on finite Riemann surfaces. Our work has its origin in a pair of theorems due to Lennart Carleson. The first of the theorems of Carleson we shall be concerned with is the following [7] : Theorem 1.1. Let fy,---,f„ be bounded holomorphic functions on U, the open unit disc, such that \fy(z) + — + \f„(z) | su <5> 0 holds for some ô and all z in U. Then there exist bounded holomorphic function gy,---,g„ on U such that ftEi + - +/A-1. In §2, we use this theorem to establish an analogous result in the setting of finite open Riemann surfaces. §§3 and 4 consider certain questions which arise naturally in the course of the proof of this generalization. We mention that the chief result of §2, Theorem 2.6, has been obtained independently by N. L. Ailing [3] who has used methods more highly algebraic than ours. The second matter we shall be concerned with is that of interpolation. If R is a Riemann surface and if £ is a subset of R, call £ an interpolation set for R if for every bounded complex-valued function a on £, there is a bounded holo- morphic function f on R such that /1 £ = a. Carleson [6] has characterized interpolation sets in the unit disc: Theorem 1.2. The set {zt}™=1of points in U is an interpolation set for U if and only if there exists ô > 0 such that for all n ** n00 k = l;ktn 1 — Z.Zl An alternative proof is to be found in [12, p.
  • Complex-Differentiability

    Complex-Differentiability

    Complex-Differentiability Sébastien Boisgérault, Mines ParisTech, under CC BY-NC-SA 4.0 April 25, 2017 Contents Core Definitions 1 Derivative and Complex-Differential 3 Calculus 5 Cauchy-Riemann Equations 7 Appendix – Terminology and Notation 10 References 11 Core Definitions Definition – Complex-Differentiability & Derivative. Let f : A ⊂ C → C. The function f is complex-differentiable at an interior point z of A if the derivative of f at z, defined as the limit of the difference quotient f(z + h) − f(z) f 0(z) = lim h→0 h exists in C. Remark – Why Interior Points? The point z is an interior point of A if ∃ r > 0, ∀ h ∈ C, |h| < r → z + h ∈ A. In the definition above, this assumption ensures that f(z + h) – and therefore the difference quotient – are well defined when |h| is (nonzero and) small enough. Therefore, the derivative of f at z is defined as the limit in “all directions at once” of the difference quotient of f at z. To question the existence of the derivative of f : A ⊂ C → C at every point of its domain, we therefore require that every point of A is an interior point, or in other words, that A is open. 1 Definition – Holomorphic Function. Let Ω be an open subset of C. A function f :Ω → C is complex-differentiable – or holomorphic – in Ω if it is complex-differentiable at every point z ∈ Ω. If additionally Ω = C, the function is entire. Examples – Elementary Functions. 1. Every constant function f : z ∈ C 7→ λ ∈ C is holomorphic as 0 λ − λ ∀ z ∈ C, f (z) = lim = 0.
  • Complex Manifolds

    Complex Manifolds

    Complex Manifolds Lecture notes based on the course by Lambertus van Geemen A.A. 2012/2013 Author: Michele Ferrari. For any improvement suggestion, please email me at: [email protected] Contents n 1 Some preliminaries about C 3 2 Basic theory of complex manifolds 6 2.1 Complex charts and atlases . 6 2.2 Holomorphic functions . 8 2.3 The complex tangent space and cotangent space . 10 2.4 Differential forms . 12 2.5 Complex submanifolds . 14 n 2.6 Submanifolds of P ............................... 16 2.6.1 Complete intersections . 18 2 3 The Weierstrass }-function; complex tori and cubics in P 21 3.1 Complex tori . 21 3.2 Elliptic functions . 22 3.3 The Weierstrass }-function . 24 3.4 Tori and cubic curves . 26 3.4.1 Addition law on cubic curves . 28 3.4.2 Isomorphisms between tori . 30 2 Chapter 1 n Some preliminaries about C We assume that the reader has some familiarity with the notion of a holomorphic function in one complex variable. We extend that notion with the following n n Definition 1.1. Let f : C ! C, U ⊆ C open with a 2 U, and let z = (z1; : : : ; zn) be n the coordinates in C . f is holomorphic in a = (a1; : : : ; an) 2 U if f has a convergent power series expansion: +1 X k1 kn f(z) = ak1;:::;kn (z1 − a1) ··· (zn − an) k1;:::;kn=0 This means, in particular, that f is holomorphic in each variable. Moreover, we define OCn (U) := ff : U ! C j f is holomorphicg m A map F = (F1;:::;Fm): U ! C is holomorphic if each Fj is holomorphic.
  • Complex Analysis Qual Sheet

    Complex Analysis Qual Sheet

    Complex Analysis Qual Sheet Robert Won \Tricks and traps. Basically all complex analysis qualifying exams are collections of tricks and traps." - Jim Agler 1 Useful facts 1 X zn 1. ez = n! n=0 1 X z2n+1 1 2. sin z = (−1)n = (eiz − e−iz) (2n + 1)! 2i n=0 1 X z2n 1 3. cos z = (−1)n = (eiz + e−iz) 2n! 2 n=0 1 4. If g is a branch of f −1 on G, then for a 2 G, g0(a) = f 0(g(a)) 5. jz ± aj2 = jzj2 ± 2Reaz + jaj2 6. If f has a pole of order m at z = a and g(z) = (z − a)mf(z), then 1 Res(f; a) = g(m−1)(a): (m − 1)! 7. The elementary factors are defined as z2 zp E (z) = (1 − z) exp z + + ··· + : p 2 p Note that elementary factors are entire and Ep(z=a) has a simple zero at z = a. 8. The factorization of sin is given by 1 Y z2 sin πz = πz 1 − : n2 n=1 9. If f(z) = (z − a)mg(z) where g(a) 6= 0, then f 0(z) m g0(z) = + : f(z) z − a g(z) 1 2 Tricks 1. If f(z) nonzero, try dividing by f(z). Otherwise, if the region is simply connected, try writing f(z) = eg(z). 2. Remember that jezj = eRez and argez = Imz. If you see a Rez anywhere, try manipulating to get ez. 3. On a similar note, for a branch of the log, log reiθ = log jrj + iθ.
  • Smooth Versus Analytic Functions

    Smooth Versus Analytic Functions

    Smooth versus Analytic functions Henry Jacobs December 6, 2009 Functions of the form X i f(x) = aix i≥0 that converge everywhere are called analytic. We see that analytic functions are equal to there Taylor expansions. Obviously all analytic functions are smooth or C∞ but not all smooth functions are analytic. For example 2 g(x) = e−1/x Has derivatives of all orders, so g ∈ C∞. This function also has a Taylor series expansion about any point. In particular the Taylor expansion about 0 is g(x) ≈ 0 + 0x + 0x2 + ... So that the Taylor series expansion does in fact converge to the function g˜(x) = 0 We see that g andg ˜ are competely different and only equal each other at a single point. So we’ve shown that g is not analytic. This is relevent in this class when finding approximations of invariant man- ifolds. Generally when we ask you to find a 2nd order approximation of the center manifold we just want you to express it as the graph of some function on an affine subspace of Rn. For example say we’re in R2 with an equilibrium point at the origin, and a center subspace along the y-axis. Than if you’re asked to find the center manifold to 2nd order you assume the manifold is locally (i.e. near (0,0)) defined by the graph (h(y), y). Where h(y) = 0, h0(y) = 0. Thus the taylor approximation is h(y) = ay2 + hot. and you must solve for a using the invariance of the manifold and the dynamics.
  • Sample Questions for Preliminary Complex Analysis Exam

    Sample Questions for Preliminary Complex Analysis Exam

    SAMPLE QUESTIONS FOR PRELIMINARY COMPLEX ANALYSIS EXAM VERSION 2.0 Contents 1. Complex numbers and functions 1 2. Definition of holomorphic function 1 3. Complex Integrals and the Cauchy Integral Formula 2 4. Sequences and series, Taylor series, and series of analytic functions 2 5. Identity Theorem 3 6. Schwarz Lemma and Cauchy Inequalities 4 7. Liouville's Theorem 4 8. Laurent series and singularities 4 9. Residue Theorem 5 10. Contour Integrals 5 11. Argument Principle 6 12. Rouch´e'sTheorem 6 13. Conformal maps 7 14. Analytic Continuation 7 15. Suggested Practice Exams 8 1. Complex numbers and functions (1.1) Write all values of ii in the form a + bi. 2 (1.2) Prove thatp sin z = z has infinitely many complex solutions. (1.3) Find log 3 + i, using the principal branch. 2. Definition of holomorphic function (2.1) Find all v : R2 ! R2 such that for z = x + iy, f(z) = (x3 − 3xy2) + iv(x; y) is analytic. (2.2) Prove that if g : C ! C is a C1 function, the following two definitions of \holomor- phic" are the same: @g (a) @z = 0 0 2 2 (b) the derivative transformation g (z0): R ! R is C-linear, for all z0 2 C. That 0 0 is, g (z0)mw = mwg (z0) for all w 2 C, where mw is the linear transformation given by complex multiplication by w. @h (2.3) True or false: If h is an entire function such that @z 6= 0 everywhere, then h is injective. 1 2 VERSION 2.0 (2.4) Find all possible a; b 2 R such that f (x; y) = x2 +iaxy +by2, x; y 2 R is holomorphic as a function of z = x + iy.
  • Chapter 2 Complex Analysis

    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.
  • Introduction to Holomorphic Functions of Several Variables, by Robert C

    Introduction to Holomorphic Functions of Several Variables, by Robert C

    BOOK REVIEWS 205 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 1, July 1991 ©1991 American Mathematical Society 0273-0979/91 $1.00+ $.25 per page Introduction to holomorphic functions of several variables, by Robert C. Gunning. Wadsworth & Brooks/Cole, vol. 1, Function the­ ory, 202 pp. ISBN 0-534-13308-8; vol. 2, Local theory, 215 pp. ISBN 0-534-13309-6; vol. 3, Homological theory, 194 pp. ISBN 0-534-13310-X What is several complex variables? The answer depends on whom you ask. Some will speak of coherent analytic sheaves, commutative al­ gebra, and Cousin problems (see for instance [GRR1, GRR2]). Some will speak of Kàhler manifolds and uniformization—how many topologically trivial complex manifolds are there of strictly negative (bounded from zero) curvature (see [KOW])? (In one complex variable the important answer is one,) Some will speak of positive line bundles (see [GRA]). Some will speak of partial differential equations, subelliptic es­ timates, and noncoercive boundary value problems (see [FOK]). Some will speak of intersection theory, orders of contact of va­ rieties with real manifolds, and algebraic geometry (see [JPDA]). Some will speak of geometric function theory (see [KRA]). Some will speak of integral operators and harmonic analysis (see [STE1, STE2, KRA]). Some will speak of questions of hard analysis inspired by results in one complex variable (see [RUD1, RUD2]). As with any lively and diverse field, there are many points of view, and many different tools that may be used to obtain useful results. Several complex variables, perhaps more than most fields, seems to be a meeting ground for an especially rich array of tech­ niques.