Real Proofs of Complex Theorems (And Vice Versa) Author(S): Lawrence Zalcman Source: the American Mathematical Monthly, Vol
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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. -
Complex Analysis
8 Complex Representations of Functions “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) 8.1 Complex Representations of Waves We have seen that we can determine the frequency content of a function f (t) defined on an interval [0, T] by looking for the Fourier coefficients in the Fourier series expansion ¥ a0 2pnt 2pnt f (t) = + ∑ an cos + bn sin . 2 n=1 T T The coefficients take forms like 2 Z T 2pnt an = f (t) cos dt. T 0 T However, trigonometric functions can be written in a complex exponen- tial form. Using Euler’s formula, which was obtained using the Maclaurin expansion of ex in Example A.36, eiq = cos q + i sin q, the complex conjugate is found by replacing i with −i to obtain e−iq = cos q − i sin q. Adding these expressions, we have 2 cos q = eiq + e−iq. Subtracting the exponentials leads to an expression for the sine function. Thus, we have the important result that sines and cosines can be written as complex exponentials: 286 partial differential equations eiq + e−iq cos q = , 2 eiq − e−iq sin q = .( 8.1) 2i So, we can write 2pnt 1 2pint − 2pint cos = (e T + e T ). -
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).......... -
Robert Macpherson Seen Through the Eyes of Others
Robert MacPherson seen through the eyes of others Compiled by Lizhen Ji June 1, 2006 Happy Birthday, Bob! 1 Introduction Robert MacPherson has made fundamental contributions to many branches of mathematics. His best known result is probably the joint work with Mark Goresky which established the intersection (co)homology of singular spaces.1 Another major piece of work, joint with Bill Fulton, also concerns the inter- section theory, which is rooted in a problem of Hilbert and led to the award2 winning book by Bill Fulton. 3 A partial list of other major topics in his work is the following: 1. Stratified Morse theory4 2. Characteristic classes of singular spaces, Rimann-Roch, K-groups etc. 3. Topological trace formula and weighted cohomology 1See Steven Kleiman’s survey paper for the discovery and many applications of the intersection homology, The development of intersection homology theory, in A century of mathematics in America, Part II, 543–585, Hist. Math., 2, Amer. Math. Soc., Providence, RI, 1989. 21996 Steele Prize of the American Mathematical Society. 3Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Springer, 1998. 4See the survey article by David Massey in the special issues of Pure and Applied Mathematics Quarterly. 1 4. Geometric approach to Langlands program (equivariant cohomology etc) 5. Reduction theory and geometry of locally symmetric varieties5 6. Kalai conjecture on IH of toric varieties 7. Combinatorial manifolds 8. Configuration spaces and compactifications of various spaces 9. Polylogarithms, etc His approach to mathematics is geometric.6 In fact, his lectures are also geometric. Pictures and diagrams in colors are an important part. -
Maa 5229 Introductory Analysis 2 Change(S) Requested
UGPC APPROVAL __________________ Florida Atlantic University UFS APPROVAL ___________________ SCNS SUBMITTAL _________________ CONFIRMED ______________________ BANNER POSTED ___________________ CATALOG POSTED __________________ Graduate Programs—COURSE CHANGE REQUEST WEB POSTED _____________________ DEPARTMENT NAME: COLLEGE OF: MATHEMATICAL SCIENCES SCIENCE COURSE PREFIX & NUMBER: CURRENT COURSE TITLE: MAA 5229 INTRODUCTORY ANALYSIS 2 CHANGE(S) REQUESTED SHOW “X” IN FRONT OF OPTION SHOW “X” IN FRONT OF OPTION CHANGE PREFIX FROM TO: CHANGE TITLE TO: CHANGE COURSE NO. FROM TO: CHANGE DESCRIPTION TO: CHANGE CREDITS FROM TO: CHANGE PREREQUISITES TO: Continuation of topics in MAA 5228. Metric space topology, uniform convergence, Arzela-Ascoli theorem, differentiation and CHANGE COREQUISITES TO: integration of single variable functions, power series, Stone- CHANGE OTHER REGISTRATION CONTROLS TO: Weierstrauss Theorem, measure theory, Lebesgue integral, convergence theorems for the Lebesgue integral, CHANGE GRADING FROM TO: absolute continuity, the fundamental theorem of calculus. OTHER CHANGES TO BE EFFECTIVE (TERM): Attach syllabus for ANY FALL 2008 changes to current course information. Will the requested change(s) cause this course to overlap any Any other departments and/or colleges that might be affected by other FAU course(s)? If yes, please list course(s). the change(s) must be consulted. List entities that have been consulted and attach written comments from each. YES NO X NONE TERMINATE COURSE, EFFECTIVE (GIVE LAST TERM COURSE IS TO BE ACTIVE): Faculty Contact, Email, Complete Phone Number: W. KALIES, WKALIES @ FAU . EDU , (561) 297-1107 SIGNATURES SUPPORTING MATERIALS Approved by: Date: Syllabus—must include all criteria as detailed in UGPC Guidelines. Department Chair: __________________________________ _________________ Go to: www. fau.edu/graduate/gpc/index.php College Curriculum Chair: ____________________________ _________________ to access Guidelines and to download this form. -
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. -
Eine Auswahl Weiterer Mathematiker, Die in Der Vorlesung Funktionentheorie Erw¨Ahnt Wurden
Eine Auswahl weiterer Mathematiker, die in der Vorlesung Funktionentheorie erw¨ahnt wurden John Wallis (* 3. Dezember 1616 in Ashford, Kent; y 8. November 1703 in Oxford) war ein englischer Mathematiker, der Beitr¨age zur Infinitesimalrechnung und zur Berechnung der Kreiszahl π leistete. Abraham de Moivre (* 26. Mai 1667 in Vitry-le-Fran¸cois; y 27. November 1754 in London) war ein franz¨osischer Mathematiker, der vor allem fur¨ den Satz von Moivre bekannt ist. James Stirling (* Mai 1692 in Garden bei Stirling; y 5. Dezember 1770 in Edinburgh) war ein schottischer Mathematiker. Joseph-Louis Lagrange (* 25. Januar 1736 in Turin als Giuseppe Lodovico Lag- rangia; y 10. April 1813 in Paris) war ein italienischer Mathematiker und Astronom. Er begrundete¨ die analytische Mechanik (Lagrangeformalismus mit der Lagrangefunktion), die er 1788 in seinem beruhmten¨ Lehrbuch M´ecaniqueanalytique\ darstellte. Weitere " Arbeitsgebiete waren das Dreik¨orperproblem der Himmelsmechanik (Lagrangepunkte), die Variationsrechnung und die Theorie der komplexen Funktionen. Lorenzo Mascheroni (* 13. Mai 1750 bei Bergamo; y 14. Juli 1800 in Paris) war ein italienischer Geistlicher und Mathematiker. Simeon´ Denis Poisson (* 21. Juni 1781 in Pithiviers (D´epartement Loiret); y 25. April 1840 in Paris) war ein franz¨osischer Physiker und Mathematiker. Augustin Jean Fresnel (* 10. Mai 1788 in Broglie (Eure); y 14. Juli 1827 in Ville- d'Avray bei Paris) war ein franz¨osischer Physiker und Ingenieur, der wesentlich zur Be- grundung¨ der Wellentheorie des Lichts und zur Optik beitrug. Augustin-Louis Cauchy (* 21. August 1789 in Paris; y 23. Mai 1857 in Sceaux) war ein franz¨osischer Mathematiker. Als ein Pionier der Analysis entwickelte er die von Gott- fried Wilhelm Leibniz und Sir Isaac Newton aufgestellten Grundlagen weiter, wo- bei er die fundamentalen Aussagen auch formal bewies. -
Mathematicians & Programmers
MATHEMATICAL SOCIETY VOLUME 9 NUMBER 6 THE AMERICAN MATHEMATICAL SOCIETY Edited by John W. Green and Gordon L. Walker CONTENTS MEETINGS Calendar of Meetings • • • • • • • • • • • . • • • . • . • . • . • • • . • . • • • • • • • 3 60 Program of the October Meeting in Hanover, New Hampshire ••.••••••• 361 Abstracts of the Meeting- pages (388-393) MEMORANDA TO MEMBERS Retired Mathematicians . • • . • . • • • • • • • • • • • • . • • • • • • • • • • • • . • • 3 63 PRELIMINARY ANNOUNCEMENTS OF MEETINGS • • • • • • • • • • • • • • • • • • • • • 364 ACTIVITIES OF OTHER ASSOCIATIONS. • • • • • • • • • • • • • • • • • • • • • • • • • • • 366 ANNUAL SALARY SURVEY •.••••••••••••••••••••••••••••••••••• 367 STARTING SALARIES FOR MATHEMATICIANS WITH A Ph.D ••••••••••••••• 370 NEWS ITEMS AND ANNOUNCEMENTS •••••••••••••••••••• 371, 374, 384 387 REPORT OF THE AFFAIRS OF THE SOCIETY •••••••••••••••••••••••• 373 PERSONAL ITEMS. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • 375 NEW AMS PUBLICATIONS. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 384 SUPPLEMENTARY PROGRAM No. 13,............................. 385 ABSTRACTS OF CONTRIBUTED PAPERS ••••••••••••.•••••••••••••• 388 INDEX TO ADVERTISERS •••••••••••••• , • • • • • • • • • • • • • • • • • • • • • • • 423 RESERVATION FORM........................................ 423 MEETINGS CALENDAR OF MEETINGS NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the NOTICES was sent to