Worked Examples in Complex Analysis
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Best Subordinant for Differential Superordinations of Harmonic Complex-Valued Functions
mathematics Article Best Subordinant for Differential Superordinations of Harmonic Complex-Valued Functions Georgia Irina Oros Department of Mathematics and Computer Sciences, Faculty of Informatics and Sciences, University of Oradea, 410087 Oradea, Romania; [email protected] or [email protected] Received: 17 September 2020; Accepted: 11 November 2020; Published: 16 November 2020 Abstract: The theory of differential subordinations has been extended from the analytic functions to the harmonic complex-valued functions in 2015. In a recent paper published in 2019, the authors have considered the dual problem of the differential subordination for the harmonic complex-valued functions and have defined the differential superordination for harmonic complex-valued functions. Finding the best subordinant of a differential superordination is among the main purposes in this research subject. In this article, conditions for a harmonic complex-valued function p to be the best subordinant of a differential superordination for harmonic complex-valued functions are given. Examples are also provided to show how the theoretical findings can be used and also to prove the connection with the results obtained in 2015. Keywords: differential subordination; differential superordination; harmonic function; analytic function; subordinant; best subordinant MSC: 30C80; 30C45 1. Introduction and Preliminaries Since Miller and Mocanu [1] (see also [2]) introduced the theory of differential subordination, this theory has inspired many researchers to produce a number of analogous notions, which are extended even to non-analytic functions, such as strong differential subordination and superordination, differential subordination for non-analytic functions, fuzzy differential subordination and fuzzy differential superordination. The notion of differential subordination was adapted to fit the harmonic complex-valued functions in the paper published by S. -
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. -
Harmonic Functions
Lecture 1 Harmonic Functions 1.1 The Definition Definition 1.1. Let Ω denote an open set in R3. A real valued function u(x, y, z) on Ω with continuous second partials is said to be harmonic if and only if the Laplacian ∆u = 0 identically on Ω. Note that the Laplacian ∆u is defined by ∂2u ∂2u ∂2u ∆u = + + . ∂x2 ∂y2 ∂z2 We can make a similar definition for an open set Ω in R2.Inthatcase, u is harmonic if and only if ∂2u ∂2u ∆u = + =0 ∂x2 ∂y2 on Ω. Some basic examples of harmonic functions are 2 2 2 3 u = x + y 2z , Ω=R , − 1 3 u = , Ω=R (0, 0, 0), r − where r = x2 + y2 + z2. Moreover, by a theorem on complex variables, the real part of an analytic function on an open set Ω in 2 is always harmonic. p R Thus a function such as u = rn cos nθ is a harmonic function on R2 since u is the real part of zn. 1 2 1.2 The Maximum Principle The basic result about harmonic functions is called the maximum principle. What the maximum principle says is this: if u is a harmonic function on Ω, and B is a closed and bounded region contained in Ω, then the max (and min) of u on B is always assumed on the boundary of B. Recall that since u is necessarily continuous on Ω, an absolute max and min on B are assumed. The max and min can also be assumed inside B, but a harmonic function cannot have any local extrema inside B. -
23. Harmonic Functions Recall Laplace's Equation
23. Harmonic functions Recall Laplace's equation ∆u = uxx = 0 ∆u = uxx + uyy = 0 ∆u = uxx + uyy + uzz = 0: Solutions to Laplace's equation are called harmonic functions. The inhomogeneous version of Laplace's equation ∆u = f is called the Poisson equation. Harmonic functions are Laplace's equation turn up in many different places in mathematics and physics. Harmonic functions in one variable are easy to describe. The general solution of uxx = 0 is u(x) = ax + b, for constants a and b. Maximum principle Let D be a connected and bounded open set in R2. Let u(x; y) be a harmonic function on D that has a continuous extension to the boundary @D of D. Then the maximum (and minimum) of u are attained on the bound- ary and if they are attained anywhere else than u is constant. Euivalently, there are two points (xm; ym) and (xM ; yM ) on the bound- ary such that u(xm; ym) ≤ u(x; y) ≤ u(xM ; yM ) for every point of D and if we have equality then u is constant. The idea of the proof is as follows. At a maximum point of u in D we must have uxx ≤ 0 and uyy ≤ 0. Most of the time one of these inequalities is strict and so 0 = uxx + uyy < 0; which is not possible. The only reason this is not a full proof is that sometimes both uxx = uyy = 0. As before, to fix this, simply perturb away from zero. Let > 0 and let v(x; y) = u(x; y) + (x2 + y2): Then ∆v = ∆u + ∆(x2 + y2) = 4 > 0: 1 Thus v has no maximum in D. -
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 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. -
Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds
Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds Xiaolong Hans Han Abstract We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hy- perbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock and Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic 3-manifolds, and formulate several questions and conjectures. Contents 1 Introduction2 1.1 Motivation and previous results . .2 1.2 A glimpse at the non-compact case . .3 1.3 Main theorem . .4 1.4 Topology and the Thurston norm of harmonic forms . .5 1.5 Minimal surfaces and the least area norm . .5 1.6 Sharpness and the interaction between harmonic forms and minimal surfaces6 1.7 Acknowledgments . .6 2 L2 Harmonic Forms and Compactly Supported Cohomology7 2.1 Basic definitions of L2 harmonic forms . .7 2.2 Hodge theory on cusped hyperbolic 3-manifolds . .9 2.3 Topology of L2-harmonic 1-forms and the Thurston Norm . 11 2.4 How much does the hyperbolic metric come into play? . 14 arXiv:2011.14457v2 [math.GT] 23 Jun 2021 3 Minimal Surface and Least Area Norm 14 3.1 Truncation of M and the definition of Mτ ................. 16 3.2 The least area norm and the L1-norm . 17 4 L1-norm, Main theorem and The Proof 18 4.1 Bounding L1-norm by L2-norm . -
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
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 IMPACT of RIEMANN's MAPPING THEOREM in the World
THE IMPACT OF RIEMANN'S MAPPING THEOREM GRANT OWEN In the world of mathematics, scholars and academics have long sought to understand the work of Bernhard Riemann. Born in a humble Ger- man home, Riemann became one of the great mathematical minds of the 19th century. Evidence of his genius is reflected in the greater mathematical community by their naming 72 different mathematical terms after him. His contributions range from mathematical topics such as trigonometric series, birational geometry of algebraic curves, and differential equations to fields in physics and philosophy [3]. One of his contributions to mathematics, the Riemann Mapping Theorem, is among his most famous and widely studied theorems. This theorem played a role in the advancement of several other topics, including Rie- mann surfaces, topology, and geometry. As a result of its widespread application, it is worth studying not only the theorem itself, but how Riemann derived it and its impact on the work of mathematicians since its publication in 1851 [3]. Before we begin to discover how he derived his famous mapping the- orem, it is important to understand how Riemann's upbringing and education prepared him to make such a contribution in the world of mathematics. Prior to enrolling in university, Riemann was educated at home by his father and a tutor before enrolling in high school. While in school, Riemann did well in all subjects, which strengthened his knowl- edge of philosophy later in life, but was exceptional in mathematics. He enrolled at the University of G¨ottingen,where he learned from some of the best mathematicians in the world at that time. -
21 Laplace's Equation and Harmonic Func- Tions
21 Laplace's Equation and Harmonic Func- tions 21.1 Introductory Remarks on the Laplacian operator Given a domain Ω ⊂ Rd, then r2u = div(grad u) = 0 in Ω (1) is Laplace's equation defined in Ω. If d = 2, in cartesian coordinates @2u @2u r2u = + @x2 @y2 or, in polar coordinates 1 @ @u 1 @2u r2u = r + : r @r @r r2 @θ2 If d = 3, in cartesian coordinates @2u @2u @2u r2u = + + @x2 @y2 @z2 while in cylindrical coordinates 1 @ @u 1 @2u @2u r2u = r + + r @r @r r2 @θ2 @z2 and in spherical coordinates 1 @ @u 1 @2u @u 1 @2u r2u = ρ2 + + cot φ + : ρ2 @ρ @ρ ρ2 @φ2 @φ sin2 φ @θ2 We deal with problems involving most of these cases within these Notes. Remark: There is not a universally accepted angle notion for the Laplacian in spherical coordinates. See Figure 1 for what θ; φ mean here. (In the rest of these Notes we will use the notation r for the radial distance from the origin, no matter what the dimension.) 1 Figure 1: Coordinate angle definitions that will be used for spherical coordi- nates in these Notes. Remark: In the math literature the Laplacian is more commonly written with the symbol ∆; that is, Laplace's equation becomes ∆u = 0. For these Notes we write the equation as is done in equation (1) above. The non-homogeneous version of Laplace's equation, namely r2u = f(x) (2) is called Poisson's equation. Another important equation that comes up in studying electromagnetic waves is Helmholtz's equation: r2u + k2u = 0 k2 is a real, positive parameter (3) Again, Poisson's equation is a non-homogeneous Laplace's equation; Helm- holtz's equation is not. -
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.