DOCSLIB.ORG

Lecture Notes on Several Complex Variables

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes on Several Complex Variables Math 650, Fall 2011 Texas A&M University Lecture notes on several complex variables Harold P. Boas Draft of November 30, 2011 Contents 1 Introduction 1 1.1 Power series . 1 1.2 Integral representations . 2 1.3 Partial differential equations . 2 1.4 Geometry . 3 2 Power series 4 2.1 Domain of convergence . 4 2.2 Characterization of domains of convergence . 5 2.3 Elementary properties of holomorphic functions . 8 2.4 Natural boundaries . 9 2.5 Summary: domains of convergence . 17 2.6 The Hartogs phenomenon . 17 2.7 Separate holomorphicity implies joint holomorphicity . 19 3 Convexity 23 3.1 Real convexity . 23 3.2 Convexity with respect to a class of functions . 24 3.2.1 Polynomial convexity . 25 3.2.2 Linear and rational convexity . 30 ii 1 Introduction Although the great Karl Weierstrass (1815–1897) studied multi-variable power series already in the nineteenth century, the modern theory of several complex variables dates to the researches of Friedrich (Fritz) Hartogs (1874–1943) in the first decade of the twentieth century.1 The so-called Hartogs Phenomenon reveals a dramatic difference between one-dimensional complex analysis and multi-dimensional complex analysis, a fundamental feature that had eluded Weierstrass. Some aspects of the theory of holomorphic (complex analytic) functions—the maximum prin- ciple, for example—are essentially the same in all dimensions. The most interesting parts of the theory of several complex variables are the features that differ from the one-dimensional theory. The one-dimensional theory is illuminated by several complementary points of view: power se- ries, integral representations, partial differential equations, and geometry. The multi-dimensional theory reveals striking new phenomena from each of these points of view. This chapter sketches some of the issues that will be treated in detail later on. 1.1 Power series A one-variable power series converges inside a certain disc and diverges outside the closure of the disc. The convergence region for a two-dimensional power series, however, can have infinitely many different shapes. For instance, the largest open set in which the double series P P n m n1 0 m1 0 ´ w converges is the unit bidisc .´; w/ ´ < 1 and w < 1 , while the series P D n Dn f W j j j j g n1 0 ´ w converges in the unbounded hyperbolic region where ´w < 1. TheD theory of one-dimensional power series bifurcates into thej theoryj of entire functions (when the series has infinite radius of convergence) and the theory of functions on the unit disc (when the series has a finite radius of convergence, which can be normalized to the value 1). In higher dimensions, studying power series already leads to function theory on infinitely many dif- ferent types of domains. A natural question, to be answered later, is to characterize the domains that are convergence domains for multi-variable power series. Exercise 1. Exhibit a two-variable power series whose convergence domain is the unit ball .´; w/ ´ 2 w 2 < 1 . f W j j C j j g 1A student of Alfred Pringsheim (1850–1941), Hartogs belonged to the Munich school of mathematicians. Because of their Jewish heritage, both Pringsheim and Hartogs suffered greatly under the Nazi regime in the 1930s. Pringsheim, a wealthy man, managed to buy his way out of Germany into Switzerland, where he died at an advanced age in 1941. The situation for Hartogs, however, grew ever more desperate, and in 1943 he committed suicide rather than face being sent to a death camp. 1 1 Introduction Hartogs discovered that every function holomorphic in a neighborhood of the boundary of the unit bidisc automatically extends to be holomorphic on the interior of the bidisc; a proof can be carried out by considering one-variable Laurent series on slices. Thus, in dramatic contrast to the situation in one variable, there are domains in C2 on which all the holomorphic functions extend to a larger domain. A natural question, to be answered later, is to characterize the domains of holomorphy, that is, the natural domains of existence of holomorphic functions. The discovery of Hartogs shows too that holomorphic functions of several variables never have isolated singularities and never have isolated zeroes, in contrast to the one-variable case. Moreover, zeroes (and singularities) must propagate to infinity or to the boundary of the domain where the function is defined. Exercise 2. Let p.´; w/ be a nonconstant polynomial in two variables. Show that the zero set of p cannot be a compact subset of C2. 1.2 Integral representations The one-variable Cauchy integral formula for a holomorphic function f on a domain bounded by a simple closed curve C says that 1 Z f .w/ f .´/ dw for ´ inside C: D 2i w ´ C 1 A remarkable feature of this formula is that the kernel .w ´/ is both universal (independent of the curve C ) and holomorphic in the free variable ´. There is no such formula in higher dimensions! There are integral representations with a holomorphic kernel that depends on the domain, and there is a universal integral representation with a kernel that is not holomorphic. There is a huge literature about constructing and analyzing integral representations for various special types of domains. 1.3 Partial differential equations The one-dimensional Cauchy–Riemann equations are a pair of real partial differential equations for a pair of functions (the real and imaginary parts of a holomorphic function). In Cn, there are still two functions, but there are 2n equations. Thus when n > 1, the inhomogeneous Cauchy– Riemann equations form an overdetermined system; hence there is a necessary compatibility condition for solvability of the Cauchy–Riemann equations. This feature is a significant differ- ence from the one-variable theory. When the inhomogeneous Cauchy–Riemann equations are solvable in C2 (or in higher dimen- sion), there is (as will be shown later) a solution with compact support in the case of compactly supported data. When n 1, however, it is not always possible to solve the inhomogeneous Cauchy–Riemann equationsD while maintaining compact support. The Hartogs phenomenon can be interpreted as a manifestation of this dimensional difference. 2 1 Introduction Exercise 3. Show that if u is the real part of a holomorphic function of two complex variables ´1 ( x1 iy1) and ´2 ( x2 iy2), then the function u must satisfy the following real second-order partialD C differential equations:D C @2u @2u @2u @2u @2u @2u @2u @2u 2 2 0; 2 2 0; 0; : @x1 C @y1 D @x2 C @y2 D @x1@x2 C @y1@y2 D @x1@y2 D @y1@x2 Thus the real part of a holomorphic function of two variables not only is harmonic in each coordinate but also satisfies additional conditions. 1.4 Geometry In view of the one-variable Riemann mapping theorem, every bounded simply connected planar domain is biholomorphically equivalent to the unit disc. In higher dimension, there is no such simple topological classification of biholomorphically equivalent domains. Indeed, the unit ball in C2 and the unit bidisc in C2 are holomorphically inequivalent domains. One way to understand intuitively why the situation changes in dimension 2 is to realize that in C2, there is room for one-dimensional complex analysis to happen in the tangent space to the boundary of a domain. Indeed, the boundary of the bidisc contains pieces of one-dimensional complex affine subspaces, while the boundary of the two-dimensional ball does not contain any such analytic disc. Similarly, the zero set of a (not identically zero) holomorphic function in C2 is a one-dimensional complex variety, while the zero set of a holomorphic function in C1 is a zero-dimensional variety (that is, a discrete set of points). There is a mismatch between the dimension of the domain and the dimension of the range of a multi-variable holomorphic function. One might expect an equidimensional holomorphic mapping to be analogous to a one-variable holomorphic function. Here too there are surprises. For instance, there exists a biholomorphic mapping from all of C2 onto a proper subset of C2 whose complement has interior points. Such a mapping is called a Fatou–Bieberbach map.2 2The name honors the French mathematician Pierre Fatou (1878–1929) and the German mathematician Ludwig Bieberbach (1886–1982). 3 2 Power series Examples in the introduction show that the domain of convergence of a multi-variable power series can have various shapes; in particular, the domain need not be a convex set. Nonetheless, there is a special kind of convexity property that characterizes convergence domains. Developing the theory requires some notation. The Cartesian product of n copies of the com- plex numbers C is denoted by Cn. In contrast to the one-dimensional case, the space Cn is not an algebra when n > 1 (there is no multiplication operation). But the space Cn is a normed vector p 2 2 space, the usual norm being the Euclidean one: .´1; : : : ; ´n/ ´1 ´n . A point n k k D j j C C j j .´1; : : : ; ´n/ in C is commonly denoted by a single letter ´, a vector variable. If ˛ is a point n ˛ ˛1 ˛n of C all of whose coordinates are non-negative integers, then ´ means the product ´1 ´n ˛1 (as usual, the quantity ´1 is interpreted as 1 when ´1 and ˛1 are simultaneously equal to 0), the notation ˛Š abbreviates the product ˛1Š ˛nŠ (where 0Š 1), and ˛ means ˛1 ˛n. In D j j C P C ˛ this “multi-index” notation, a multi-variable power series can be written in the form ˛ c˛´ , P P ˛1 ˛n 1 1 an abbreviation for ˛1 0 ˛n 0 c˛1;:::;˛n ´1 ´n .
Load more

Recommended publications
  • Real Proofs of Complex Theorems (And Vice Versa)
    REAL PROOFS OF COMPLEX THEOREMS (AND VICE VERSA) LAWRENCE ZALCMAN Introduction. It has become fashionable recently to argue that real and complex variables should be taught together as a unified curriculum in analysis. Now this is hardly a novel idea, as a quick perusal of Whittaker and Watson's Course of Modern Analysis or either Littlewood's or Titchmarsh's Theory of Functions (not to mention any number of cours d'analyse of the nineteenth or twentieth century) will indicate. And, while some persuasive arguments can be advanced in favor of this approach, it is by no means obvious that the advantages outweigh the disadvantages or, for that matter, that a unified treatment offers any substantial benefit to the student. What is obvious is that the two subjects do interact, and interact substantially, often in a surprising fashion. These points of tangency present an instructor the opportunity to pose (and answer) natural and important questions on basic material by applying real analysis to complex function theory, and vice versa. This article is devoted to several such applications. My own experience in teaching suggests that the subject matter discussed below is particularly well-suited for presentation in a year-long first graduate course in complex analysis. While most of this material is (perhaps by definition) well known to the experts, it is not, unfortunately, a part of the common culture of professional mathematicians. In fact, several of the examples arose in response to questions from friends and colleagues. The mathematics involved is too pretty to be the private preserve of specialists.
    [Show full text]
  • Robert De Montessus De Ballore's 1902 Theorem on Algebraic
    Robert de Montessus de Ballore’s 1902 theorem on algebraic continued fractions : genesis and circulation Hervé Le Ferrand ∗ October 31, 2018 Abstract Robert de Montessus de Ballore proved in 1902 his famous theorem on the convergence of Padé approximants of meromorphic functions. In this paper, we will first describe the genesis of the theorem, then investigate its circulation. A number of letters addressed to Robert de Montessus by different mathematicians will be quoted to help determining the scientific context and the steps that led to the result. In particular, excerpts of the correspondence with Henri Padé in the years 1901-1902 played a leading role. The large number of authors who mentioned the theorem soon after its derivation, for instance Nörlund and Perron among others, indicates a fast circulation due to factors that will be thoroughly explained. Key words Robert de Montessus, circulation of a theorem, algebraic continued fractions, Padé’s approximants. MSC : 01A55 ; 01A60 1 Introduction This paper aims to study the genesis and circulation of the theorem on convergence of algebraic continued fractions proven by the French mathematician Robert de Montessus de Ballore (1870-1937) in 1902. The main issue is the following : which factors played a role in the elaboration then the use of this new result ? Inspired by the study of Sturm’s theorem by Hourya Sinaceur [52], the scientific context of Robert de Montessus’ research will be described. Additionally, the correlation with the other topics on which he worked will be highlighted,
    [Show full text]
  • European Influences in the Fine Arts: Melbourne 1940-1960
    INTERSECTING CULTURES European Influences in the Fine Arts: Melbourne 1940-1960 Sheridan Palmer Bull Submitted in total fulfilment of the requirements of the degree ofDoctor ofPhilosophy December 2004 School of Art History, Cinema, Classics and Archaeology and The Australian Centre The University ofMelbourne Produced on acid-free paper. Abstract The development of modern European scholarship and art, more marked.in Austria and Germany, had produced by the early part of the twentieth century challenging innovations in art and the principles of art historical scholarship. Art history, in its quest to explicate the connections between art and mind, time and place, became a discipline that combined or connected various fields of enquiry to other historical moments. Hitler's accession to power in 1933 resulted in a major diaspora of Europeans, mostly German Jews, and one of the most critical dispersions of intellectuals ever recorded. Their relocation to many western countries, including Australia, resulted in major intellectual and cultural developments within those societies. By investigating selected case studies, this research illuminates the important contributions made by these individuals to the academic and cultural studies in Melbourne. Dr Ursula Hoff, a German art scholar, exiled from Hamburg, arrived in Melbourne via London in December 1939. After a brief period as a secretary at the Women's College at the University of Melbourne, she became the first qualified art historian to work within an Australian state gallery as well as one of the foundation lecturers at the School of Fine Arts at the University of Melbourne. While her legacy at the National Gallery of Victoria rests mostly on an internationally recognised Department of Prints and Drawings, her concern and dedication extended to the Gallery as a whole.
    [Show full text]
  • Arxiv:1803.01386V4 [Math.HO] 25 Jun 2021
    2009 SEKI http://wirth.bplaced.net/seki.html ISSN 1860-5931 arXiv:1803.01386v4 [math.HO] 25 Jun 2021 A Most Interesting Draft for Hilbert and Bernays’ “Grundlagen der Mathematik” that never found its way into any publi- Working-Paper cation, and 2 CVof Gisbert Hasenjaeger Claus-Peter Wirth Dept. of Computer Sci., Saarland Univ., 66123 Saarbrücken, Germany [email protected] SEKI Working-Paper SWP–2017–01 SEKI SEKI is published by the following institutions: German Research Center for Artificial Intelligence (DFKI GmbH), Germany • Robert Hooke Str.5, D–28359 Bremen • Trippstadter Str. 122, D–67663 Kaiserslautern • Campus D 3 2, D–66123 Saarbrücken Jacobs University Bremen, School of Engineering & Science, Campus Ring 1, D–28759 Bremen, Germany Universität des Saarlandes, FR 6.2 Informatik, Campus, D–66123 Saarbrücken, Germany SEKI Editor: Claus-Peter Wirth E-mail: [email protected] WWW: http://wirth.bplaced.net Please send surface mail exclusively to: DFKI Bremen GmbH Safe and Secure Cognitive Systems Cartesium Enrique Schmidt Str. 5 D–28359 Bremen Germany This SEKI Working-Paper was internally reviewed by: Wilfried Sieg, Carnegie Mellon Univ., Dept. of Philosophy Baker Hall 161, 5000 Forbes Avenue Pittsburgh, PA 15213 E-mail: [email protected] WWW: https://www.cmu.edu/dietrich/philosophy/people/faculty/sieg.html A Most Interesting Draft for Hilbert and Bernays’ “Grundlagen der Mathematik” that never found its way into any publication, and two CV of Gisbert Hasenjaeger Claus-Peter Wirth Dept. of Computer Sci., Saarland Univ., 66123 Saarbrücken, Germany [email protected] First Published: March 4, 2018 Thoroughly rev. & largely extd. (title, §§ 2, 3, and 4, CV, Bibliography, &c.): Jan.
    [Show full text]
  • Pringsheim: Maiolica
    Seven pieces of maiolica from the former Pringsheim Collection, inv. nos. A 3582, A 3602, A 3619, A 3638, A 3648, A 3649, A 3653 >ORIGINAL DUTCH VERSION In 2008 the Museum Boijmans Van Beuningen Foundation received a letter written on behalf of the heirs of the German collector Prof. Alfred Pringsheim (1850-1941), the owner of a celebrated collection of Italian maiolica, from which seven pieces were acquired by the collector J.N. Bastert that are now in Museum Boijmans Van Beuningen. The heirs asked the Museum Boijmans Van Beuningen Foundation, which was the owner of the objects, to return them. The museum responded by proposing that the question be jointly submitted for a binding recommendation to the Advisory Committee on the Assessment for Items of Cultural Value and the Second World War (the Restitutions Committee). The museum is currently awaiting the heirs’ reply, but in the meantime has launched its own investigation of the history of the collection, which is being carried out by the art historian Judith Niessen. Alfred Pringsheim was a well-known and highly respected professor of mathematics at the Ludwig-Maximilian University in Munich. His father left him a large fortune made in the coal and railway industries, which enabled Pringsheim to become a major benefactor. He and his wife Hedwig Dohm, a prominent feminist who was the daughter of a journalist, entertained many guests from the cultural elite of Bavaria and Munich in the ‘palace’ they had built in Narcissstrasse. Thomas Mann, one of their regular visitors, married Katia, the Pringsheim’s daughter, who bore him five children.
    [Show full text]
  • Bibliographic Essay for Alex Ross's Wagnerism: Art and Politics in The
    Bibliographic Essay for Alex Ross’s Wagnerism: Art and Politics in the Shadow of Music The notes in the printed text of Wagnerism give sources for material quoted in the book and cite the important primary and secondary literature on which I drew. From those notes, I have assembled an alphabetized bibliography of works cited. However, my reading and research went well beyond the literature catalogued in the notes, and in the following essay I hope to give as complete an accounting of my research as I can manage. Perhaps the document will be of use to scholars doing further work on the phenomenon of Wagnerism. As I indicate in my introduction and acknowledgments, I am tremendously grateful to those who have gone before me; a not inconsiderable number of them volunteered personal assistance as I worked. Wagner has been the subject of thousands of books—although the often-quoted claim that more has been written about him than anyone except Christ or Napoleon is one of many indestructible Wagner myths. (Barry Millington, long established one of the leading Wagner commentators in English, disposes of it briskly in an essay on “Myths and Legends” in his Wagner Compendium, published by Schirmer in 1992.) Nonetheless, the literature is vast, and since Wagner himself is not the central focus of my book I won’t attempt any sort of broad survey here. I will, however, indicate the major works that guided me in assembling the piecemeal portrait of Wagner that emerges in my book. The most extensive biography, though by no means the most trustworthy, is the six- volume, thirty-one-hundred-page life by the Wagner idolater Carl Friedrich Glasenapp (Breitkopf und Härtel, 1894–1911).
    [Show full text]
  • Robert Lehman Papers
    Robert Lehman papers Finding aid prepared by Larry Weimer The Robert Lehman Collection Archival Project was generously funded by the Robert Lehman Foundation, Inc. This finding aid was generated using Archivists' Toolkit on September 24, 2014 Robert Lehman Collection The Metropolitan Museum of Art 1000 Fifth Avenue New York, NY, 10028 [email protected] Robert Lehman papers Table of Contents Summary Information .......................................................................................................3 Biographical/Historical note................................................................................................4 Scope and Contents note...................................................................................................34 Arrangement note.............................................................................................................. 36 Administrative Information ............................................................................................ 37 Related Materials ............................................................................................................ 39 Controlled Access Headings............................................................................................. 41 Bibliography...................................................................................................................... 40 Collection Inventory..........................................................................................................43 Series I. General
    [Show full text]
  • Badw · 019750249
    BAYERISCHE AKADEMIE DER WISSENSCHAFTEN MATHEMATISCH-NATURWISSENSCHAFTLICHE KLASSE SITZUNGSBERICHTE JAHRGANG 1997 MÜNCHEN 1997 VERLAG DER BAYERISCHEN AKADEMIE DER WISSENSCHAFTEN In Kommission bei der C. H. Beck’schen Verlagsbuchhandlung München Pringsheim, Liebmann, Hartogs - Schicksale jüdischer Mathematiker in München von Friedrich L. Bauer Sitzung vom 21. Februar 1997 Die Liste deutscher Mathematiker, die von den Nationalsoziali- sten verfolgt wurden, ist lang. Viele darunter wurden wegen ihrer jüdischen Rasse“ beleidigt, gequält, ihrer Menschenwürde be- raubt, ermordet oder in den Tod getrieben. Der bereits in den zwanziger Jahren des Jahrhunderts deutlich ausgeprägte Antisemi- tismus veranlaßte nur wenige, rechtzeitig auszuwandem: Adolph Abraham Fraenkel (1891—1965) ging 1929 an die hebräische Uni- versität in Jerusalem, Felix Bernstein verließ Göttingen im De- zember 1932, John von Neumann (1903—1957), der in Budapest promoviert hatte und 1926 als Rockefeller Fellow nach Göttingen kam (er war 1927-1929 Privatdozent in Berlin und 1929/1930 Privatdozent in Flamburg) war seit 1930 Gastdozent an der Prin- ceton University und nahm im Januar 1933 eine Stellung am In- stitute for Advanced Studies in Princeton an. Der junge Hans Ar- nold Heilbronn (1908—1975), Landaus Assistent, ging 1933 nach England. Wolfgang Doeblin (1915-1940), Sohn des Schriftstellers Alfred Doeblin, emigrierte ebenfalls 1933, wurde französischer Staatsbürger und fiel 1940 für Frankreich. Die Verfolgung beginnt Am 7. April 1933 wurde das ,Gesetz zur Wiederherstellung des Berufsbeamtentums“ verkündet, in dessen § 3 es hieß: „Beamte, die nicht-arischer Abstammung sind, sind in den Ruhestand [...] zu versetzen“ und dessen § 4 verfugte „Beamte, die nach ihrer bishe- rigen politischen Betätigung nicht die Gewähr dafür bieten, daß sie jederzeit rückhaltslos für den nationalen Staat eintreten, können aus dem Dienst entlassen werden“.
    [Show full text]
  • Acceptance of Western Piano Music in Japan and the Career of Takahiro Sonoda
    UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE THE ACCEPTANCE OF WESTERN PIANO MUSIC IN JAPAN AND THE CAREER OF TAKAHIRO SONODA A DOCUMENT SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the Degree of DOCTOR OF MUSICAL ARTS By MARI IIDA Norman, Oklahoma 2009 © Copyright by MARI IIDA 2009 All Rights Reserved. ACKNOWLEDGEMENTS My document has benefitted considerably from the expertise and assistance of many individuals in Japan. I am grateful for this opportunity to thank Mrs. Haruko Sonoda for requesting that I write on Takahiro Sonoda and for generously providing me with historical and invaluable information on Sonoda from the time of the document’s inception. I must acknowledge my gratitude to following musicians and professors, who willingly told of their memories of Mr. Takahiro Sonoda, including pianists Atsuko Jinzai, Ikuko Endo, Yukiko Higami, Rika Miyatani, Y ōsuke Niin ō, Violinist Teiko Maehashi, Conductor Heiichiro Ōyama, Professors Jun Ozawa (Bunkyo Gakuin University), Sh ūji Tanaka (Kobe Women’s College). I would like to express my gratitude to Teruhisa Murakami (Chief Concert Engineer of Yamaha), Takashi Sakurai (Recording Engineer, Tone Meister), Fumiko Kond ō (Editor, Shunju-sha) and Atsushi Ōnuki (Kajimoto Music Management), who offered their expertise to facilitate my understanding of the world of piano concerts, recordings, and publications. Thanks are also due to Mineko Ejiri, Masako Ōhashi for supplying precious details on Sonoda’s teaching. A special debt of gratitude is owed to Naoko Kato in Tokyo for her friendship, encouragement, and constant aid from the beginning of my student life in Oklahoma. I must express my deepest thanks to Dr.
    [Show full text]
  • Constantin Carathéodory, a Greek-German Professor Of
    Online-Publication as personal statement, June 1, 2017 Constantin Carath´eodory, a Greek-German Professor of Mathematics at the University of Munich in the Years 1924–1945 — A Personal Statement Hans Josef Pesch Abstract The present paper reflects the life of the Greek-German mathemati- cian Constantin Carath´eodory as a professor of mathematics at the University of Munich from the year of his call in 1924 over the terrible era of the National Socialist dictatorship until the end of World War II. Although Carath´eodory’s biography seems to be widely known, at least among mathematicians, little seems to be known of his life during the pandemonium of darkness that stran- gled millions in the German name. Two more or less recent publications seem to provide points of attack for casting a cloud, resp. actually do cast a cloud over Carath´eodory. The present paper discusses these criticisms in terms of the history of the backdoor acquisition of power by the Nazis and the backdrop of day-to-day life outside and inside the university system at that time. The available historical sources, in particular those of the mentioned publications, disburden Carath´eodory in the opinion of the author and lead to the conclusion that he must have had a hostile attitude toward the NS regime despite his decision to accept the call from the University of Munich in 1924, at a time, when nationalistic and xenophobic assaults began to occur more and more, and despite his decision to reject possibilities to emigrate to the US or Greece before 1933.
    [Show full text]
  • Complex Numbers and Colors
    Complex Numbers and Colors This year’s calendar, “Complex Beauties,” delights both the visual senses and the intellect. These colorful representations of complex functions are fascinating aesthetically as well as mathematically. The researchers’ biographical sketches and the explanations of the mathematical functions with which their work is connected provides insight into various fields of mathematics. Our goal is to inspire you to search the literature or to experiment with phase portraits on your own. To understand the representations of these functions, it is helpful to know something about com- plex numbers. The construction of the phase portraits is based on the interpretation of complex numbers z as points in the Gaussian plane. The horizontal coordinate x of the point representing z is called the real part of z (Re z) and the vertical coordinate y of the point representing z is called the imaginary part of z (Im z); we write z = x + iy. Alternatively, the point representing z can also be given by its distance from the origin ( z , the modulus of z) and an angle ( arg z, the argument of z). | | The phase portrait of a complex function f (appearing in the picture on the left) arises when all points z of the domain of f are colored according to the argument (or phase) of the value w = f (z). More precisely, in the first step the color wheel is used to assign colors to the complex w-plane in such a way that points on rays emanating from the origin are assigned the same color (as in the picture on the right).
    [Show full text]
  • The Transcendence of Pi Has Been Known for About a Century
    Results in Mathematics Contents 117 J. Bair/F. Jongmans Volume 7/No. 2/1984 Some remarks about recent result Pages 117-250 on the asymptotic cone 119 W. Beekmann/S. C Chang On the structure of summability fields 130 P. Bundschuh/1. Shiokawa A measure for the linear inde- pendence of certain numbers 145 P. L. Butzer/R. J. Nessel/E. L. Stark Eduard Helly (1884-1943) in memoriam 154 A. S. Cavaretta jr./H. P. Dikshit/ A. Sharma An extension of a theorem of Walsh 164 R. Fritsch The transcendence of ir has been known for about a century-but who was the man who discovered it? 184 Y. Hirano/H. Tominaga On simple ring extensions gener- ated by two idempotents 190 J. Joussen Eine Bemerkung zu einem Satz von Sylvester 192 H. Karzel/C. J. Maxson Fibered groups with non-trivial centers 209 H. Meiert G. Rosenberger Hecke-Integrale mit rationalen periodischen Funktionen und Dirichlet-Reihen mit Funk• tionalgleichung 234 G. Schiffels/M. Stemel Einbettung von topologischen Ringen in Quotientenringe Short Communications on Mathematical Dissertations 249 G. BaszenskijW. Schempp TAXT Konvergenzbeschleunigung von Orthogonal-Doppelreihen The Journal Copyright RESULTS IN MATHEMATICS It is a fundamental condition of publication that submitted RESULTATE DER MATHEMATIK manuscripts have not been published, nor will be simulta- publishes mainly research papers in all Heids of pure and applied mathematics. In addition, it publishes summaries of any mathematical neously submitted or published elsewhere. By submitting a field and surveys of any mathematical subject provided they are de- manuscript, the authors agree that the Copyright for their signed to advance some recent mathematical development.
    [Show full text]