Harmonic Function Theory
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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. -
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. -
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 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. -
A Comparison of Viola Strings with Harmonic Frequency Analysis
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Student Research, Creative Activity, and Performance - School of Music Music, School of 5-2011 A Comparison of Viola Strings with Harmonic Frequency Analysis Jonathan Paul Crosmer University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/musicstudent Part of the Music Commons Crosmer, Jonathan Paul, "A Comparison of Viola Strings with Harmonic Frequency Analysis" (2011). Student Research, Creative Activity, and Performance - School of Music. 33. https://digitalcommons.unl.edu/musicstudent/33 This Article is brought to you for free and open access by the Music, School of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Student Research, Creative Activity, and Performance - School of Music by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. A COMPARISON OF VIOLA STRINGS WITH HARMONIC FREQUENCY ANALYSIS by Jonathan P. Crosmer A DOCTORAL DOCUMENT Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Musical Arts Major: Music Under the Supervision of Professor Clark E. Potter Lincoln, Nebraska May, 2011 A COMPARISON OF VIOLA STRINGS WITH HARMONIC FREQUENCY ANALYSIS Jonathan P. Crosmer, D.M.A. University of Nebraska, 2011 Adviser: Clark E. Potter Many brands of viola strings are available today. Different materials used result in varying timbres. This study compares 12 popular brands of strings. Each set of strings was tested and recorded on four violas. We allowed two weeks after installation for each string set to settle, and we were careful to control as many factors as possible in the recording process. -
Musical Acoustics - Wikipedia, the Free Encyclopedia 11/07/13 17:28 Musical Acoustics from Wikipedia, the Free Encyclopedia
Musical acoustics - Wikipedia, the free encyclopedia 11/07/13 17:28 Musical acoustics From Wikipedia, the free encyclopedia Musical acoustics or music acoustics is the branch of acoustics concerned with researching and describing the physics of music – how sounds employed as music work. Examples of areas of study are the function of musical instruments, the human voice (the physics of speech and singing), computer analysis of melody, and in the clinical use of music in music therapy. Contents 1 Methods and fields of study 2 Physical aspects 3 Subjective aspects 4 Pitch ranges of musical instruments 5 Harmonics, partials, and overtones 6 Harmonics and non-linearities 7 Harmony 8 Scales 9 See also 10 External links Methods and fields of study Frequency range of music Frequency analysis Computer analysis of musical structure Synthesis of musical sounds Music cognition, based on physics (also known as psychoacoustics) Physical aspects Whenever two different pitches are played at the same time, their sound waves interact with each other – the highs and lows in the air pressure reinforce each other to produce a different sound wave. As a result, any given sound wave which is more complicated than a sine wave can be modelled by many different sine waves of the appropriate frequencies and amplitudes (a frequency spectrum). In humans the hearing apparatus (composed of the ears and brain) can usually isolate these tones and hear them distinctly. When two or more tones are played at once, a variation of air pressure at the ear "contains" the pitches of each, and the ear and/or brain isolate and decode them into distinct tones. -
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. -
Harmonic and Real Analysis Herbert Koch Universit¨Atbonn Wintersemester 2014-2015
Notes for Harmonic and real analysis Herbert Koch Universit¨atBonn Wintersemester 2014-2015 Recommended literature: [10, 7, 14, 13, 15] 1 Contents Chapter 1. Introduction 5 1. An example 5 Chapter 2. Fourier series 9 1. Definitions 9 2. The convolution and Young's inequality 10 3. Lp convergence of partial sums 15 4. Regularity and Fourier series 16 5. Complex Interpolation 17 6. The Hardy-Littlewood maximal function and real interpolation 21 7. Higher dimensions 30 Chapter 3. Harmonic functions and the Poisson kernel 31 1. Basic properties and the Poisson kernel 31 2. Boundary behaviour of harmonic functions 36 3. Almost everywhere convergence 37 4. Subharmonic functions 38 5. The theorems of Riesz 40 6. Conjugate functions 41 7. The Hilbert transform 43 n Chapter 4. The Fourier transform on R 47 1. Definition and first properties 47 2. The Fourier transform of Schwartz functions 48 3. The Fourier transform on tempered distributions 51 4. Oscillatory integrals and stationary phase 58 5. Quantization 64 6. Cotlar's Lemma and the Theorem of Calder´on-Vaillancourt 70 Chapter 5. Singular integrals of Calder´on-Zygmund type 75 1. The setting 75 2. The Calder´on-Zygmund theorem 77 3. Examples 81 Chapter 6. Hardy and BMO 87 1. More general maximal functions and the Hardy space Hp 87 2. The atomic decomposition 91 3. Duality and BMO 95 4. Relation to harmonic functions 98 5. Div-curl type results 100 6. Application to elliptic PDEs 101 7. Pointwise estimates and perturbations of elliptic equations 107 3 4 CONTENTS Chapter 7. -
Harmonic Functions
6 HARMONIC FUNCTIONS This chapter is devoted to the Laplace equation. We introduce two of its important properties, the maximum principle and the rotational invariance. Then we solve the equation in series form in rectangles, circles, and related shapes. The case of a circle leads to the beautiful Poisson formula. 6.1 LAPLACE’S EQUATION If a diffusion or wave process is stationary (independent of time), then ut 0 ≡ and utt 0. Therefore, both the diffusion and the wave equations reduce to the Laplace≡ equation: uxx 0 in one dimension = u u uxx u yy 0 in two dimensions ∇·∇ = = + = u u uxx u yy uzz 0 in three dimensions ∇·∇ = = + + = A solution of the Laplace equation is called a harmonic function. In one dimension, we have simply uxx 0, so the only harmonic functions in one dimension are u(x) A Bx. But=this is so simple that it hardly gives us a clue to what happens =in higher+ dimensions. The inhomogeneous version of Laplace’s equation u f (1) = with f a given function, is called Poisson’s equation. Besides stationary diffusions and waves, some other instances of Laplace’s and Poisson’s equations include the following. 152 6.1 LAPLACE’S EQUATION 153 1. Electrostatics. From Maxwell’s equations, one has curl E 0 and div E 4πρ, where ρ is the charge density. The first equation implies= E grad= φ for a scalar function φ (called the electric potential). Therefore,=− φ div(grad φ) div E 4πρ , = =− =− which is Poisson’s equation (with f 4πρ ). =− 2. Steady fluid flow.