A Cauchy-Dirac Delta Function

The Dirac δ function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac’s discovery by over a century, and illuminating the nature of Cauchy’s infinitesimals and his infinitesimal definition of δ.

[1]  J. Fourier.  Théorie analytique de la chaleur , 2009 .

[2]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[3]  Joseph W Dauben,et al.  Abraham Robinson and nonstandard analysis: history, philosophy, and foundations of mathematics , 1988 .

[4]  Mikhail G. Katz,et al.  From discrete arithmetic to arithmetic of the continuum , 2013 .

[5]  R. Ely.  Nonstandard Student Conceptions About Infinitesimals , 2010 .

[6]  Abraham Robinson,et al.  Nonstandard analysis and philosophy , 1979 .

[7]  H. Keisler.  Elementary Calculus: An Infinitesimal Approach , 1976 .

[8]  Kajsa Bråting,et al.  A new look at E.G. Björling and the Cauchy sum theorem , 2007 .

[9]  Edward Nelson.  Internal set theory: A new approach to nonstandard analysis , 1977 .

[10]  Alexandre Borovik,et al.  Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus , 2011, 1108.2885.

[11]  Jerzy Loś,et al.  Quelques Remarques, Théorèmes Et Problèmes Sur Les Classes Définissables D'algèbres , 1955 .

[12]  D. Laugwitz.  Early delta functions and the use of infinitesimals in research , 1992 .

[13]  Philip Ehrlich,et al.  The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes , 2006 .

[14]  Alexandre V. Borovik,et al.  An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals , 2012, Notre Dame J. Formal Log..

[15]  Mikhail G. Katz,et al.  Cauchy's Continuum , 2011, Perspectives on Science.

[16]  Mikhail G. Katz,et al.  Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond , 2012, 1205.0174.

[17]  David Salinger.  Plato's Ghost: The Modernist Transformation of Mathematics , 2010 .

[18]  The Prehistory of the Theory of Distributions. By Jesper Lützen , 1984 .

[19]  Abraham Robinson.  Selected papers of Abraham Robinson , 1978 .

[20]  A. Robinson.  Non-standard analysis , 1966 .

[21]  Karin U. Katz,et al.  When is .999... less than 1? , 2010, The Mathematics Enthusiast.

[22]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .

[23]  Detlef Laugwitz.  Infinitely small quantities in Cauchy's textbooks , 1987 .

[24]  Stephen W. Hawking,et al.  God Created the Integers: The Mathematical Breakthroughs That Changed History , 2005 .

[25]  John N. Crossley,et al.  Problems in the Philosophy of Mathematics , 1968, The Mathematical Gazette.

[26]  The Metaphysics of the Calculus , 1967 .

[27]  Ekkehard Kopp,et al.  On Cauchy's Notion of Infinitesimal , 1988, The British Journal for the Philosophy of Science.

[28]  D. Tall,et al.  THE TENSION BETWEEN INTUITIVE INFINITESIMALS AND FORMAL MATHEMATICAL ANALYSIS , 2011, 1110.5747.

[29]  Mikhail G. Katz,et al.  Zooming in on infinitesimal 1–.9.. in a post-triumvirate era , 2010, 1003.1501.

[30]  P. Zsombor-Murray,et al.  Elementary Mathematics from an Advanced Standpoint , 1940, Nature.

[31]  Edwin Hewitt,et al.  Rings of real-valued continuous functions. I , 1948 .

[32]  M. Katz,et al.  Two ways of obtaining infinitesimals by refining Cantor's completion of the reals , 2011, 1109.3553.

[33]  D. Laugwitz,et al.  Eine Erweiterung der Infinitesimalrechnung , 1958 .

[34]  Mikhail G. Katz,et al.  Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond , 2012, Perspectives on Science.

[35]  T. Skolem.  Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen , 1934 .

[36]  Jérôme Havenel.  Peirce's Clarifications of Continuity , 2008 .

[37]  Leif Arkeryd,et al.  Intermolecular forces of infinite range and the Boltzmann equation , 1981 .

[38]  M. Barany.  God, king, and geometry: revisiting the introduction to Cauchys Cours danalyse , 2011 .

[39]  G. Fisher,et al.  The infinite and infinitesimal quantities of du Bois-Reymond and their reception , 1981 .

[40]  C. Gilain.  Cauchy et le cours d'analyse de l'Ecole polytechnique , 1989 .

[41]  A. Cauchy.  Cours d'analyse de l'École royale polytechnique , 1821 .

[42]  Augustin-Louis Cauchy.  Oeuvres complètes: Mémoire sur les développements des fonctions en séries périodiques , 2009 .

[43]  Y. Péraire.  A mathematical framework for Dirac's calculus , 2007 .

[44]  Mikhail G. Katz,et al.  Meaning in Classical Mathematics: Is it at Odds with Intuitionism? , 2011, 1110.5456.

[45]  E. Perkins.  NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS , 1988 .

[46]  Abraham Adolf Fraenkel.  Einleitung in die Mengenlehre , 1919 .

[47]  Augustin-Louis Cauchy.  Oeuvres complètes: ANALYSE MATHÉMATIQUE. — Note sur les séries convergentes dont les divers termes sont des fonctions continues d'une variable réelle ou imaginaire, entre des limites données , 2009 .

[48]  D. Laugwitz.  Definite values of infinite sums: Aspects of the foundations of infinitesimal analysis around 1820 , 1989 .

[49]  K. D. Stroyan.  Uniform Continuity and Rates of Growth of Meromorphic Functions1) , 1972 .

[50]  E. Seneta.  Cauchy, Augustin–Louis , 2006 .

[51]  David Sherry,et al.  The wake of Berkeley's analyst: Rigor mathematicae? , 1987 .

[52]  A. Robinson.  Numbers and Models , Standard and Nonstandard , 2010 .

[53]  The correctness of Euler's method for the factorization of the sine function into an infinite product , 1988 .

[54]  Mikhail G. Katz,et al.  Leibniz's laws of continuity and homogeneity , 2012, 1211.7188.

[55]  Mikhail G. Katz,et al.  Ten Misconceptions from the History of Analysis and Their Debunking , 2012, 1202.4153.

[56]  Hideyasu Yamashita.  Pointwise analysis of scalar fields: A nonstandard approach , 2006 .

[57]  Todor D. Todorov,et al.  A nonstandard delta function , 1990 .

[58]  Carl B. Boyer.  The concepts of the calculus : a critical and historical discussion of the derivative and the integral , 1939 .

[59]  Mikhail G. Katz,et al.  A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography , 2011, 1104.0375.