Introduction to Complex Analysis
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Lecture 5: Complex Logarithm and Trigonometric Functions
LECTURE 5: COMPLEX LOGARITHM AND TRIGONOMETRIC FUNCTIONS Let C∗ = C \{0}. Recall that exp : C → C∗ is surjective (onto), that is, given w ∈ C∗ with w = ρ(cos φ + i sin φ), ρ = |w|, φ = Arg w we have ez = w where z = ln ρ + iφ (ln stands for the real log) Since exponential is not injective (one one) it does not make sense to talk about the inverse of this function. However, we also know that exp : H → C∗ is bijective. So, what is the inverse of this function? Well, that is the logarithm. We start with a general definition Definition 1. For z ∈ C∗ we define log z = ln |z| + i argz. Here ln |z| stands for the real logarithm of |z|. Since argz = Argz + 2kπ, k ∈ Z it follows that log z is not well defined as a function (it is multivalued), which is something we find difficult to handle. It is time for another definition. Definition 2. For z ∈ C∗ the principal value of the logarithm is defined as Log z = ln |z| + i Argz. Thus the connection between the two definitions is Log z + 2kπ = log z for some k ∈ Z. Also note that Log : C∗ → H is well defined (now it is single valued). Remark: We have the following observations to make, (1) If z 6= 0 then eLog z = eln |z|+i Argz = z (What about Log (ez)?). (2) Suppose x is a positive real number then Log x = ln x + i Argx = ln x (for positive real numbers we do not get anything new). -
Branch Points and Cuts in the Complex Plane
BRANCH POINTS AND CUTS IN THE COMPLEX PLANE Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog and include the title or URL of this post in your comment. Post date: 6 July 2021. We’ve looked at contour integration in the complex plane as a technique for evaluating integrals of complex functions and finding infinite integrals of real functions. In some cases, the complex functions that are to be integrated are multi- valued. As a preliminary to the contour integration of such functions, we’ll look at the concepts of branch points and branch cuts here. The stereotypical function that is used to introduce branch cuts in most books is the complex logarithm function logz which is defined so that elogz = z (1) If z is real and positive, this reduces to the familiar real logarithm func- tion. (Here I’m using natural logs, so the real natural log function is usually written as ln. In complex analysis, the term log is usually used, so be careful not to confuse it with base 10 logs.) To generalize it to complex numbers, we write z in modulus-argument form z = reiθ (2) and apply the usual rules for taking a log of products and exponentials: logz = logr + iθ (3) = logr + iargz (4) To see where problems arise, suppose we start with z on the positive real axis and increase θ. Everything is fine until θ approaches 2π. When θ passes 2π, the original complex number z returns to its starting value, as given by 2. -
Riemann Surfaces
RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. -
3 Elementary Functions
3 Elementary Functions We already know a great deal about polynomials and rational functions: these are analytic on their entire domains. We have thought a little about the square-root function and seen some difficulties. The remaining elementary functions are the exponential, logarithmic and trigonometric functions. 3.1 The Exponential and Logarithmic Functions (§30–32, 34) We have already defined the exponential function exp : C ! C : z 7! ez using Euler’s formula ez := ex cos y + iex sin y (∗) and seen that its real and imaginary parts satisfy the Cauchy–Riemann equations on C, whence exp C d z = z is entire (analytic on ). Indeed recall that dz e e . We have also seen several of the basic properties of the exponential function, we state these and several others for reference. Lemma 3.1. Throughout let z, w 2 C. 1. ez 6= 0. ez 2. ez+w = ezew and ez−w = ew 3. For all n 2 Z, (ez)n = enz. 4. ez is periodic with period 2pi. Indeed more is true: ez = ew () z − w = 2pin for some n 2 Z Proof. Part 1 follows trivially from (∗). To prove 2, recall the multiple-angle formulae for cosine and sine. Part 3 requires an induction using part 2 with z = w. Part 4 is more interesting: certainly ew+2pin = ew by the periodicity of sine and cosine. Now suppose ez = ew where z = x + iy and w = u + iv. Then, by considering the modulus and argument, ( ex = eu exeiy = eueiv =) y = v + 2pin for some n 2 Z We conclude that x = u and so z − w = i(y − v) = 2pin. -
Complex Analysis
Complex Analysis Andrew Kobin Fall 2010 Contents Contents Contents 0 Introduction 1 1 The Complex Plane 2 1.1 A Formal View of Complex Numbers . .2 1.2 Properties of Complex Numbers . .4 1.3 Subsets of the Complex Plane . .5 2 Complex-Valued Functions 7 2.1 Functions and Limits . .7 2.2 Infinite Series . 10 2.3 Exponential and Logarithmic Functions . 11 2.4 Trigonometric Functions . 14 3 Calculus in the Complex Plane 16 3.1 Line Integrals . 16 3.2 Differentiability . 19 3.3 Power Series . 23 3.4 Cauchy's Theorem . 25 3.5 Cauchy's Integral Formula . 27 3.6 Analytic Functions . 30 3.7 Harmonic Functions . 33 3.8 The Maximum Principle . 36 4 Meromorphic Functions and Singularities 37 4.1 Laurent Series . 37 4.2 Isolated Singularities . 40 4.3 The Residue Theorem . 42 4.4 Some Fourier Analysis . 45 4.5 The Argument Principle . 46 5 Complex Mappings 47 5.1 M¨obiusTransformations . 47 5.2 Conformal Mappings . 47 5.3 The Riemann Mapping Theorem . 47 6 Riemann Surfaces 48 6.1 Holomorphic and Meromorphic Maps . 48 6.2 Covering Spaces . 52 7 Elliptic Functions 55 7.1 Elliptic Functions . 55 7.2 Elliptic Curves . 61 7.3 The Classical Jacobian . 67 7.4 Jacobians of Higher Genus Curves . 72 i 0 Introduction 0 Introduction These notes come from a semester course on complex analysis taught by Dr. Richard Carmichael at Wake Forest University during the fall of 2010. The main topics covered include Complex numbers and their properties Complex-valued functions Line integrals Derivatives and power series Cauchy's Integral Formula Singularities and the Residue Theorem The primary reference for the course and throughout these notes is Fisher's Complex Vari- ables, 2nd edition. -
The Monodromy Groups of Schwarzian Equations on Closed
Annals of Mathematics The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces Author(s): Daniel Gallo, Michael Kapovich and Albert Marden Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 151, No. 2 (Mar., 2000), pp. 625-704 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/121044 . Accessed: 15/02/2013 18:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded on Fri, 15 Feb 2013 18:57:11 PM All use subject to JSTOR Terms and Conditions Annals of Mathematics, 151 (2000), 625-704 The monodromy groups of Schwarzian equations on closed Riemann surfaces By DANIEL GALLO, MICHAEL KAPOVICH, and ALBERT MARDEN To the memory of Lars V. Ahlfors Abstract Let 0: 7 (R) -* PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for 0 to be the monodromy represen- tation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that 0(7ri(R)) be nonelementary. -
Inverse Trigonometric Functions
Chapter 2 INVERSE TRIGONOMETRIC FUNCTIONS vMathematics, in general, is fundamentally the science of self-evident things. — FELIX KLEIN v 2.1 Introduction In Chapter 1, we have studied that the inverse of a function f, denoted by f–1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses. In Class XI, we studied that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist. In this chapter, we shall study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behaviour through graphical representations. Besides, some elementary properties will also be discussed. The inverse trigonometric functions play an important Aryabhata role in calculus for they serve to define many integrals. (476-550 A. D.) The concepts of inverse trigonometric functions is also used in science and engineering. 2.2 Basic Concepts In Class XI, we have studied trigonometric functions, which are defined as follows: sine function, i.e., sine : R → [– 1, 1] cosine function, i.e., cos : R → [– 1, 1] π tangent function, i.e., tan : R – { x : x = (2n + 1) , n ∈ Z} → R 2 cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R π secant function, i.e., sec : R – { x : x = (2n + 1) , n ∈ Z} → R – (– 1, 1) 2 cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1) 2021-22 34 MATHEMATICS We have also learnt in Chapter 1 that if f : X→Y such that f(x) = y is one-one and onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y. -
Complex Analysis
COMPLEX ANALYSIS MARCO M. PELOSO Contents 1. Holomorphic functions 1 1.1. The complex numbers and power series 1 1.2. Holomorphic functions 3 1.3. Exercises 7 2. Complex integration and Cauchy's theorem 9 2.1. Cauchy's theorem for a rectangle 12 2.2. Cauchy's theorem in a disk 13 2.3. Cauchy's formula 15 2.4. Exercises 17 3. Examples of holomorphic functions 18 3.1. Power series 18 3.2. The complex logarithm 20 3.3. The binomial series 23 3.4. Exercises 23 4. Consequences of Cauchy's integral formula 25 4.1. Expansion of a holomorphic function in Taylor series 25 4.2. Further consequences of Cauchy's integral formula 26 4.3. The identity principle 29 4.4. The open mapping theorem and the principle of maximum modulus 30 4.5. The general form of Cauchy's theorem 31 4.6. Exercises 36 5. Isolated singularities of holomorphic functions 37 5.1. The residue theorem 40 5.2. The Riemann sphere 42 5.3. Evalutation of definite integrals 43 5.4. The argument principle and Rouch´e's theorem 48 5.5. Consequences of Rouch´e'stheorem 49 5.6. Exercises 51 6. Conformal mappings 53 6.1. Fractional linear transformations 56 6.2. The Riemann mapping theorem 58 6.3. Exercises 63 7. Harmonic functions 65 7.1. Maximum principle 65 7.2. The Dirichlet problem 68 7.3. Exercises 71 8. Entire functions 72 8.1. Infinite products 72 Appunti per il corso Analisi Complessa per i Corsi di Laurea in Matematica dell'Universit`adi Milano. -
Complex Numbers and Functions
Complex Numbers and Functions Richard Crew January 20, 2018 This is a brief review of the basic facts of complex numbers, intended for students in my section of MAP 4305/5304. I will discuss basic facts of com- plex arithmetic, limits and derivatives of complex functions, power series and functions like the complex exponential, sine and cosine which can be defined by convergent power series. This is a preliminary version and will be added to later. 1 Complex Numbers 1.1 Arithmetic. A complex number is an expression a + bi where i2 = −1. Here the real number a is the real part of the complex number and bi is the imaginary part. If z is a complex number we write <(z) and =(z) for the real and imaginary parts respectively. Two complex numbers are equal if and only if their real and imaginary parts are equal. In particular a + bi = 0 only when a = b = 0. The set of complex numbers is denoted by C. Complex numbers are added, subtracted and multiplied according to the usual rules of algebra: (a + bi) + (c + di) = (a + c) + (b + di) (1.1) (a + bi) − (c + di) = (a − c) + (b − di) (1.2) (a + bi)(c + di) = (ac − bd) + (ad + bc)i (1.3) (note how i2 = −1 has been used in the last equation). Division performed by rationalizing the denominator: a + bi (a + bi)(c − di) (ac − bd) + (bc − ad)i = = (1.4) c + di (c + di)(c − di) c2 + d2 Note that denominator only vanishes if c + di = 0, so that a complex number can be divided by any nonzero complex number. -
Universit`A Degli Studi Di Perugia the Lambert W Function on Matrices
Universita` degli Studi di Perugia Facolta` di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Triennale in Informatica The Lambert W function on matrices Candidato Relatore MassimilianoFasi BrunoIannazzo Contents Preface iii 1 The Lambert W function 1 1.1 Definitions............................. 1 1.2 Branches.............................. 2 1.3 Seriesexpansions ......................... 10 1.3.1 Taylor series and the Lagrange Inversion Theorem. 10 1.3.2 Asymptoticexpansions. 13 2 Lambert W function for scalar values 15 2.1 Iterativeroot-findingmethods. 16 2.1.1 Newton’smethod. 17 2.1.2 Halley’smethod . 18 2.1.3 K¨onig’s family of iterative methods . 20 2.2 Computing W ........................... 22 2.2.1 Choiceoftheinitialvalue . 23 2.2.2 Iteration.......................... 26 3 Lambert W function for matrices 29 3.1 Iterativeroot-findingmethods. 29 3.1.1 Newton’smethod. 31 3.2 Computing W ........................... 34 3.2.1 Computing W (A)trougheigenvectors . 34 3.2.2 Computing W (A) trough an iterative method . 36 A Complex numbers 45 A.1 Definitionandrepresentations. 45 B Functions of matrices 47 B.1 Definitions............................. 47 i ii CONTENTS C Source code 51 C.1 mixW(<branch>, <argument>) ................. 51 C.2 blockW(<branch>, <argument>, <guess>) .......... 52 C.3 matW(<branch>, <argument>) ................. 53 Preface Main aim of the present work was learning something about a not- so-widely known special function, that we will formally call Lambert W function. This function has many useful applications, although its presence goes sometimes unrecognised, in mathematics and in physics as well, and we found some of them very curious and amusing. One of the strangest situation in which it comes out is in writing in a simpler form the function . -
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. -
An Example of a Covering Surface with Movable Natural Boundaries
An example of a covering surface with movable natural boundaries Claudi Meneghin November 13, 2018 Abstract We exhibit a covering surface of the punctured complex plane (with no points over 0) whose natural projection mapping fails to be a topo- logical covering, due to the existence of branches with natural bound- ∗ aries, projecting over different slits in C . In [1], A.F.Beardon shows an example of a ’covering surface’ over a region in the unit disc which is not a topological covering (We recall that a ’covering surface’ of a region D ⊂ C is a Riemann surface S admitting a surjective conformal mapping onto D, among which there are the analytical continua- tions of holomorphic germs; a continuous mapping p of a topological space Y onto another one X is a ’topological covering’ provided that each point x ∈ X admits an open neighbourhood U such that the restriction of p to −1 each connected component Vi of p (U) is a homeomorphism of Vi onto U, see [2, 3]). Beardon’s example takes origin by the analytical continuation of a holo- morphic germ, namely a branch of the inverse of the Blaschke product arXiv:math/0408086v3 [math.CV] 11 Jul 2008 ∞ z − a B(z)= n , n 1 − anz Y=1 with suitable hypotheses about the an’s. It is shown that there is a sequence of inverse branches fk of B at 0 such that the radii of convergence of their Taylor developments tend to 0 as k →∞. This of course prevents the natural projection (taking a germ at z0 to the point z0) of the analytical continuation of any branch of B−1 to be a topological covering.