Complex Analysis
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HOMOTECIA Nº 6-15 Junio 2017
HOMOTECIA Nº 6 – Año 15 Martes, 1º de Junio de 2017 1 Entre las expectativas futuras que se tienen sobre un docente en formación, está el considerar como indicativo de que logrará realizarse como tal, cuando evidencia confianza en lo que hace, cuando cree en sí mismo y no deja que su tiempo transcurra sin pro pósitos y sin significado. Estos son los principios que deberán pautar el ejercicio de su magisterio si aspira tener éxito en su labor, lo cual mostrará mediante su afán por dar lo bueno dentro de sí, por hacer lo mejor posible, por comprometerse con el porvenir de quienes confiadamente pondrán en sus manos la misión de enseñarles. Pero la responsabilidad implícita en este proceso lo debería llevar a considerar seriamente algunos GIACINTO MORERA (1856 – 1907 ) aspectos. Obtener una acreditación para enseñar no es un pergamino para exhib ir con petulancia ante familiares y Nació el 18 de julio de 1856 en Novara, y murió el 8 de febrero de 1907, en Turín; amistades. En otras palabras, viviendo en el mundo educativo, es ambas localidades en Italia. asumir que se produjo un cambio significativo en la manera de Matemático que hizo contribuciones a la dinámica. participar en este: pasó de ser guiado para ahora guiar. No es que no necesite que se le orie nte como profesional de la docencia, esto es algo que sucederá obligatoriamente a nivel organizacional, Giacinto Morera , hijo de un acaudalado hombre de pero el hecho es que adquirirá una responsabilidad mucho mayor negocios, se graduó en ingeniería y matemáticas en la porque así como sus preceptores universitarios tuvieron el compromiso de formarlo y const ruirlo cultural y Universidad de Turín, Italia, habiendo asistido a los académicamente, él tendrá el mismo compromiso de hacerlo con cursos por Enrico D'Ovidio, Angelo Genocchi y sus discípulos, sea cual sea el nivel docente donde se desempeñe. -
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. -
4 Complex Analysis
4 Complex Analysis “He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.” Henry David Thoreau (1817 - 1862) We have seen that we can seek the frequency content of a signal f (t) defined on an interval [0, T] by looking for the the Fourier coefficients in the Fourier series expansion In this chapter we introduce complex numbers and complex functions. We a ¥ 2pnt 2pnt will later see that the rich structure of f (t) = 0 + a cos + b sin . 2 ∑ n T n T complex functions will lead to a deeper n=1 understanding of analysis, interesting techniques for computing integrals, and The coefficients can be written as integrals such as a natural way to express analog and dis- crete signals. 2 Z T 2pnt an = f (t) cos dt. T 0 T However, we have also seen that, using Euler’s Formula, trigonometric func- tions can be written in a complex exponential form, 2pnt e2pint/T + e−2pint/T cos = . T 2 We can use these ideas to rewrite the trigonometric Fourier series as a sum over complex exponentials in the form ¥ 2pint/T f (t) = ∑ cne , n=−¥ where the Fourier coefficients now take the form Z T −2pint/T cn = f (t)e dt. -
Sunti Delle Conferenze
Sunti delle Conferenze Analisi complessa a Pisa, 1860-1900 UMBERTO BOTTAZZINI (Università di Milano) Nel 1859 Enrico Betti inaugura gli studi di analisi complessa a Pisa (e di fatto in Italia) pubblicando la traduzione italiana della Inauguraldissertation (1851) di Riemann. L’incontro con il grande matematico conosciuto l’anno prima a Göttingen segna una svolta nella carriera scientifica di Betti, che fa dell’analisi complessa l’oggetto delle sue lezioni e delle sue pubblicazioni (1860/61 e 1862) che incontrano l’approvazione di Riemann, durante il suo soggiorno in Italia. Nella conferenza saranno discussi i contributi all’analisi complessa di Betti, Dini e Bianchi. Ulisse Dini raccolse l’eredità del maestro dapprima in articoli (1870/71, 1871/73, 1881) che suscitano l’interesse della comunità internazionale, e poi in lezioni litografate (1890) che hanno offerto a Luigi Bianchi il modello e il riferimento iniziale per le sue celebri lezioni sulla teoria delle funzioni di variabile complessa in due volumi, apparse prima in versione litografata (1898/99) e poi a stampa in diverse edizioni. Il periodo romano di Luigi Cremona: tra Statica Grafica e Geometria Algebrica, la Biblioteca Nazionale, i Lincei, il Senato ALDO BRIGAGLIA (Università di Palermo) Il periodo romano (1873 – 1903) è considerato il meno produttivo, dal punto di vista scientifico, della vita di Luigi Cremona. Un periodo quasi unicamente dedicato agli aspetti politico – istituzionali della sua attività. Senza voler capovolgere questo giudizio consolidato, anzi sottolineando -
Considerations Based on His Correspondence Rossana Tazzioli
New Perspectives on Beltrami’s Life and Work - Considerations Based on his Correspondence Rossana Tazzioli To cite this version: Rossana Tazzioli. New Perspectives on Beltrami’s Life and Work - Considerations Based on his Correspondence. Mathematicians in Bologna, 1861-1960, 2012, 10.1007/978-3-0348-0227-7_21. hal- 01436965 HAL Id: hal-01436965 https://hal.archives-ouvertes.fr/hal-01436965 Submitted on 22 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. New Perspectives on Beltrami’s Life and Work – Considerations based on his Correspondence Rossana Tazzioli U.F.R. de Math´ematiques, Laboratoire Paul Painlev´eU.M.R. CNRS 8524 Universit´ede Sciences et Technologie Lille 1 e-mail:rossana.tazzioliatuniv–lille1.fr Abstract Eugenio Beltrami was a prominent figure of 19th century Italian mathematics. He was also involved in the social, cultural and political events of his country. This paper aims at throwing fresh light on some aspects of Beltrami’s life and work by using his personal correspondence. Unfortunately, Beltrami’s Archive has never been found, and only letters by Beltrami - or in a few cases some drafts addressed to him - are available. -
Mathematik Für Physiker
Mathematik f¨urPhysiker III Michael Dreher Fachbereich f¨urMathematik und Statistik Universit¨atKonstanz Studienjahr 2012/13 2 Acknowledgements: These are the lecture notes to a third semester course on Mathematics for Physicists, and the author is indebted to Nicola Wurz, Maria Lindauer, Florian Franz, Philip Lindner, Simon Sch¨uz,Samuel Greiner, Christian Schoder, Pascal Gumbsheimer for remarks which helped to improve the presentation, and to the audience for appropriating this huge amount of knowledge. Some Legalese: This work is licensed under the Creative Commons Attribution { Noncommercial { No Derivative Works 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea{shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Sir Isaac Newton (1642 { 1727) 1 1As quoted in [3]. 4 Contents I Ordinary Differential Equations7 1 Existence and Uniqueness Results9 1.1 An Introductory Example....................................9 1.2 General Considerations...................................... 12 1.3 The Theorem of Picard and Lindelof¨ ............................ 19 1.4 Comparison Principles...................................... 23 2 Special Solution Methods 25 2.1 Equations with Separable Variables............................... 25 2.2 Substitution and Homogeneous Differential Equations.................... 27 2.3 Power Series Expansions (Or How to Determine the Sound of a Drum).......... -
Arxiv:1911.12155V13 [Math.CA] 1 Jul 2020 2.7 Double Integration
On logarithmic integrals, harmonic sums and variations Ming Hao Zhao Abstract Based on various non-MZV approaches we evaluate certain logarithmic integrals and har- monic sums. More specifically, 85 LIs, 89 ESs, 263 PLIs, 28 non-alt QESs, 39 GESs, 26 BESs, 14 IBESs with weight ≤ 5, 193 QLIs, 172 QPLIs, 83 QESs, 7 QBESs, 3 IQBESs with weight ≤ 4. With help of MZV theory we evaluate other 10 QESs with weight ≤ 4. High weight hypergeometric series, nonhomogeneous integrals and other results are also established. Contents 1 Preliminaries 5 1.1 Special functions . .5 1.2 Definition of subjects . .6 1.2.1 LI and PLI . .6 1.2.2 QLI and QPLI . .7 1.2.3 ES and QES . .8 1.2.4 Fibonacci basis . .9 1.3 Main results . 11 2 LI 11 2.1 Brute force . 11 2.2 General formulas . 12 2.3 Integration by parts . 14 2.4 Fractional transformation . 14 2.5 Beta derivatives . 15 2.6 Contour integration . 16 arXiv:1911.12155v13 [math.CA] 1 Jul 2020 2.7 Double integration . 16 2.8 Hypergeometric identities . 16 2.9 Solution . 17 1 3 ES 17 3.1 General formulas . 17 3.2 Symmetric relations . 18 3.3 Partial fractions . 18 3.4 Contour integration . 18 3.5 Solution . 18 3.5.1 Preparation . 18 3.5.2 Confrontation . 24 4 PLI 27 4.1 Brute force . 27 4.2 General formulas . 27 4.3 Integration by parts . 28 4.4 Fractional transformation . 28 4.5 Complementary integrals . 28 4.6 Polylog identities . 28 4.7 Series expansion . -
Integration and Summation
Integration and Summation Edward Jin Contents 1 Preface 3 2 Riemann Sums 4 2.1 Theory . .4 2.2 Exercises . .5 2.3 Solutions . .6 3 Weierstrass Substitutions 7 3.1 Theory . .7 3.2 Exercises . .8 3.3 Solutions . .9 4 Clever Substitutions 10 4.1 Theory . 10 4.2 Exercises . 11 4.2.1 Solutions . 12 5 Integral and Summation Substitutions 13 5.1 Theory . 13 5.2 Exercises . 15 5.3 Solutions . 16 6 Substitutions About the Domain 17 6.1 Theory . 17 6.2 Exercises . 19 6.3 Solutions . 20 7 Gaussian Integral; Gamma/Beta Functions 21 7.1 Gamma Function . 21 1 CONTENTS 2 7.2 Beta Function . 21 7.3 Gaussian Integral . 22 7.4 Exercises . 24 7.5 Solutions . 25 8 Differentiation Under the Integral Sign 27 8.1 Theory . 27 8.2 Exercises . 28 8.3 Solutions . 29 9 Frullani’s Theorem 30 9.1 Exercises . 30 9.2 Solutions . 31 10 Further Exercises 32 Chapter 1 Preface This text is designed to introduce various techniques in Integration and Summation, which are commonly seen in Integration Bees and other such contests. The text is designed to be accessible to those who have completed a standard single-variable calculus course. Examples, Exercises, and Solutions are presented in each section in order to help the reader become become acquainted with the techniques presented. It is assumed that the reader is familiar with single-variable calculus methods of integration, including u-substitution, integration by parts, trigonometric substitution, and partial fractions. It is also assumed that the reader is familiar with trigonometric and logarithmic identities. -
Gli Archivi Di Corrado Segre Presso L'università Di Torino
Gli Archivi di Corrado Segre presso l’Università di Torino LIVIA GIACARDI, ERIKA LUCIANO, CHIARA PIZZARELLI, CLARA SILVIA ROERO Alla scomparsa di Corrado Segre nel Maggio 1924, sua moglie Olga Michelli donò alla Biblioteca Speciale di Matematica della Facoltà di Scienze Matematiche Fisiche e Naturali dell’Università di Torino la sua raccolta di opuscoli, estratti e ritratti1, completando il lascito nel Marzo 1926, con la donazione dei celebri quaranta Quaderni delle sue lezioni e con una collezione di manoscritti e documenti scientifici del marito 2. In occasione delle Celebrazioni per il 150° anniversario della nascita di Segre3, nel 2013 Paola Gario (Università Statale di Milano) consegnò alla medesima Biblioteca Speciale di Matematica trentaquattro tavole di Geometria Proiettiva e Descrittiva, che Segre aveva realizzato nel 1878-79, all’epoca dei suoi studi superiori presso l’Istituto tecnico ‘G. Sommeiller’ di Torino. Gario aveva ricevuto tali tavole dagli eredi di Segre nel 1989. Nel Gennaio 2014, e successivamente nell’ottobre 2015, l’Università di Torino ricevette la donazione di una ricca raccolta di tavole, corrispondenze, necrologi e documenti di differente tipologia, già custoditi ad Ancona dai discendenti di Segre: i pronipoti Silvano e Daniele Fuà4. Si fornisce qui una preliminare ricognizione del complesso di questi fondi d’archivio, posticipandone l’analisi storiografica alla conclusione dell’opera di catalogazione e di schedatura. Gli Archivi Segre possono essere articolati nelle seguenti dieci Serie. UTo-ACS Università di Torino, Archivi di Corrado Segre I. Documenti di Carriera - Corrispondenza istituzionale - Attestati e lettere di Accademie e Società scientifiche - Premi di studio, diplomi e onorificenze II. Documenti di Famiglia - Indirizzario di Segre - Lettere di Segre a O. -
Superior Mathematics from an Elementary Point of View ” Course Notes Jacopo D’Aurizio
” Superior Mathematics from an Elementary point of view ” course notes Jacopo d’Aurizio To cite this version: Jacopo d’Aurizio. ” Superior Mathematics from an Elementary point of view ” course notes. Master. Italy. 2017. cel-01813978 HAL Id: cel-01813978 https://cel.archives-ouvertes.fr/cel-01813978 Submitted on 12 Jun 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. \Superior Mathematics from an Elementary point of view" course notes Undergraduate course, 2017-2018, University of Pisa Jack D'Aurizio Contents 0 Introduction 2 1 Creative Telescoping and DFT 3 2 Convolutions and ballot problems 15 3 Chebyshev and Legendre polynomials 30 4 The glory of Fourier, Laplace, Feynman and Frullani 40 5 The Basel problem 60 6 Special functions and special products 70 7 The Cauchy-Schwarz inequality and beyond 97 8 Remarkable results in Linear Algebra 121 9 The Fundamental Theorem of Algebra 125 10 Quantitative forms of the Weierstrass approximation Theorem 133 11 Elliptic integrals and the AGM 137 12 Dilworth, Erdos-Szekeres, Brouwer and Borsuk-Ulam's Theorems 147 13 Continued fractions and elements of Diophantine Approximation 158 14 Symmetric functions and elements of Analytic Combinatorics 173 15 Spherical Trigonometry 183 0 Introduction This course has been designed to serve University students of the first and second year of Mathematics. -
Real Proofs of Complex Theorems (And Vice Versa) Author(S): Lawrence Zalcman Source: the American Mathematical Monthly, Vol
Real Proofs of Complex Theorems (and Vice Versa) Author(s): Lawrence Zalcman Source: The American Mathematical Monthly, Vol. 81, No. 2 (Feb., 1974), pp. 115-137 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2976953 Accessed: 05-04-2017 21:47 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/2976953?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. 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]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly This content downloaded from 108.179.181.50 on Wed, 05 Apr 2017 21:47:20 UTC All use subject to http://about.jstor.org/terms 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. -
The Weierstrass Substitution
Academic Forum 30 2012-13 Cauchy Confers with Weierstrass Lloyd Edgar S. Moyo, Ph.D. Associate Professor of Mathematics Abstract. We point out two limitations of using the Cauchy Residue Theorem to evaluate a definite integral of a real rational function of cos θ and sin θ . We also exhibit a definite integral with the limitations. Finally, we overcome the limitations by using the Weierstrass substitution. Introduction Let R be a rational function of two real variables. Consider a real integral 2π ∫ R(cos θ,sin θ)dθ . 0 By using the substitution z = e iθ , where i= −1 and θ is a real number, it can be shown that 2π R(cos θ,sin θ)dθ = − i 1R(z 2 +1 , z 2 −1 )dz . (1) ∫ ∫ |=z| 1 z 2z 2iz 0 For details, see, for example, Palka [4], p. 329 or Ahlfors [1], p. 154 or Crowder and McCuskey [2], p. 258. By the Cauchy Residue Theorem, we have n n 1 z 2 +1 z 2 −1 − i R( , )dz = (−i)( 2πi) Re s( f (z); zk ) = 2π Re s( f (z); zk ,) ∫ |=z| 1 z 2z 2iz ∑ ∑ k =1 k =1 where 1 z2 +1 z2 −1 f (z) = z R( 2z , 2iz ), z1,..., zn are the poles of f in the open unit disk D = {z ∈C:|z|<1}, and Res ( f(z); zk ) denotes the residue of f at the pole zk . Hence, (1) becomes 2π n ∫ R(cos θ,sin θ)dθ = 2π ∑ Re s( f (z); zk .) (2) 0 k =1 Limitations of Formula (2) In this section, we point out two cases for which the formula (2) does not apply and we suggest a different method of attack.