The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass

Bottazzini, Umberto

doi:10.1007/978-1-4612-4944-3

Cited by 179 publications

(43 citation statements)

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“…Relativamente ao primeiro item, indicamos a relevância do entendimento do processo matemático de integração, desde os seus rudimentos na variável real e, alguns séculos depois, a busca pela sua generalidade e consistência das estruturas construídas em que foram estabelecidas os fundamentos da teoria das funções na variável complexa (BOTTAZZINI, 1986;BOTTAZZINI & GRAY, 2013;GRAY, 2015;DEBNATH, 2015;HAIRER & WANNER, 2008;MEDVEDEV, 1991;SCHUBRING, 2005 …”

Section: Análises Preliminaresunclassified

Engenharia Didática Com O Tema Integração De Funções Na Variável Complexa: Análises Preliminares, a Priori E Modelização De Situações

Alves¹

2017

Ens. C. e Tec. em Rev.

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Resumo: O presente escrito aborda, discute e caracteriza as fases iniciais de um design de investigação e pesquisa em Didática da Matemática, com um tema envolvendo o processo matemático de integração de funções na variável complexa. Dessa forma, após a indicação de fatores preocupantes no âmbito da Transição Complexa do Cálculo -TCC, bem como a identificação de elementos de transição e elementos de ruptura no âmbito do seu ensino, apresenta elementos fundamentais que consubstanciam a etapa de análise preliminar e análise a priori e, logo em seguida, a concepção de situações-problema. Desse modo, ao assumir um papel distinguido para a visualização como componente preliminar para a promoção de um entendimento tácito e intuitivo para a noção simbolizada de modo standard portrabalho fornece e acentua um conjunto de elementos que não podem ser desconsiderados numa transposição didática que se apoia na tecnologia atual e se mostra inspirada pela Teoria das Situações Didáticas -TSD. Palavras-chave: Engenharia Didática, Integração de Funções, Análise Complexa, Visualização.Abstract: This writting addresses, discusses and caracterizes the early stages of a research design in Didactics of Mathematics, with a theme involving the mathematical processs of integration of complex variable functions. Thus, after the indications of worrying factors within the Transition of Complex Calculus -TCC, and the identification of some transition and breaking elements within the teaching context, presents fundamental elements that embody the preliminary analysis stage and the a priori analysis and, shortly thereafter, the construction of some problem of situations. Thus, to take a dintinguided conception of the visualization as preliminar components for promoting an tacit and intuitive understanding for the standard symbolized notions as, the work provides and enhances a numbers of elements that can not be disregarded in a didactical transposition that it is based on current technology and shows be inspired in Theory of Didatical Situations -TDS.

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Section: Análises Preliminaresunclassified

Engenharia Didática Com O Tema Integração De Funções Na Variável Complexa: Análises Preliminares, a Priori E Modelização De Situações

Alves¹

2017

Ens. C. e Tec. em Rev.

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“…In particular, Weierstrass's 'Epsilontik' (so called for the usage of and δ as variable names) would become the standard approach to analysis (cf. e.g., Bottazzini, 1986). Even if other non-mainstream approaches were present in the mathematical literature at the time (cf.…”

Section: Hatchet Jobs: Cohen's Book and Its Reviewersmentioning

confidence: 99%

Hermann Cohen's Das Princip der Infinitesimal-Methode: The history of an unsuccessful book

Giovanelli¹

2016

Studies in History and Philosophy of Science Part A

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This paper offers an introduction to Hermann Cohen's Das Princip der Infinitesimal-Methode (1883), and recounts the history of its controversial reception by Cohen's early sympathizers, who would become the so-called 'Marburg school' of Neo-Kantianism, as well as the reactions it provoked outside this group. By dissecting the ambiguous attitudes of the best-known representatives of the school (Paul Natorp and Ernst Cassirer), as well as those of several minor figures (August Stadler, Kurd Lasswitz, Dimitry Gawronsky, etc.), this paper shows that Das Princip der Infinitesimal-Methode is a unicum in the history of philosophy: it represents a strange case of an unsuccessful book's enduring influence. The "puzzle of Cohen's Infinitesimalmethode," as we will call it, can be solved by looking beyond the scholarly results of the book, and instead focusing on the style of philosophy it exemplified. Moreover, the paper shows that Cohen never supported, but instead explicitly opposed, the doctrine of the centrality of the 'concept of function', with which Marburg Neo-Kantianism is usually associated.

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“…12 Let {x n } n∈Z ⊂ X be a sequence in a distance space (X, d). The sequence {x n } is complete in (X, d) if the following implication holds: {x n } is Cauchy in (X, d) =⇒ {x n } is convergent in (X, d).…”

Section: Definitionsmentioning

confidence: 99%

Properties of distance spaces with power triangle inequalities

Greenhoe

2016

Carpathian Math. Publ.

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Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.

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The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass

Cited by 179 publications

References 0 publications

Engenharia Didática Com O Tema Integração De Funções Na Variável Complexa: Análises Preliminares, a Priori E Modelização De Situações

Engenharia Didática Com O Tema Integração De Funções Na Variável Complexa: Análises Preliminares, a Priori E Modelização De Situações

Hermann Cohen's Das Princip der Infinitesimal-Methode: The history of an unsuccessful book

Properties of distance spaces with power triangle inequalities

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