@article{bwmeta1.element.bwnjournal-article-bcpv53z1p11bwm, author = {Synowiec, John}, title = {Some highlights in the development of algebraic analysis}, journal = {Banach Center Publications}, volume = {51}, year = {2000}, pages = {11-46}, zbl = {1074.01517}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv53z1p11bwm} }
Synowiec, John. Some highlights in the development of algebraic analysis. Banach Center Publications, Tome 51 (2000) pp. 11-46. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv53z1p11bwm/
[000] L.-F. A. ARBOGAST [1789], Essai sur de nouveaux principes de Calcul différentiel et intégral indépendans de la théorie des infiniment-petits et celle de limites, unpublished manuscript.
[001] L.-F. A. ARBOGAST [1800], Du calcul des dérivations, Strasbourg, 404 pp.
[002] C. BABBAGE [1827], On the influence of signs in mathematical reasoning, Trans. Cambridge Phil. Soc. Vol. II, Part II, 325-377.
[003] S. BANACH [1939], Über das 'loi suprème' von Hoene-Wroński, Bull. Acad. Polon. Sci. et Lettres Sér. A 1-10. | Zbl 65.1318.01
[004] F. A. BEREZIN [1987], Introduction to Superanalysis, D. Reidel, 424 pp.
[005] R. BITTNER [1964], Algebraic and analytic properties of solutions of abstract differential equations, Rozprawy Matem. 41, 63 pp. % | Zbl 0191.15004
[006] J.-E. BJÖRK [1979], Rings of Differential Operators, North-Holland, 374 pp.
[007] S. BOCHNER [1932], Vorlesungen über Fouriersche Integrale, Leipzig, 229 pp. | Zbl 58.0292.01
[008] G. BOOLE [1844], On a general method in analysis, Phil. Trans. Roy. Soc. London 134, 225-282.
[009] G. BOOLE [1847], On a certain symbolical equation, Cambridge & Dublin Math. J. 2, 7-12.
[010] G. BOOLE [1859], A Treatise on Differential Equations, Macmillan, London, 485 pp. (1877, 566 pp.)
[011] G. BOOLE [1860], A Treatise on the Calculus of Finite Differences, Macmillan, London, 420 pp., 3rd ed., 1880, 336 pp. | Zbl 1202.65003
[012] U. BOTTAZZINI [1986], The Higher Calculus, Springer, 332 pp. | Zbl 0597.01011
[013] N. BOURBAKI [1994], Elements of the History of Mathematics, Springer, 301 pp. | Zbl 0803.01002
[014] T. J. I'A. BROMWICH [1916], Normal coordinates in dynamical systems, Proc. London Math. Soc. 15, 401-448. | Zbl 46.1181.05
[015] F. BRACKX, R. DELANGHE & F. SOMMEN [1982], Clifford Analysis, Pitman, 308 pp.
[016] B. BRISSON [1808], Mémoire sur l'intégration des équations différentielles partielles, J. de l'École Polytech. 7, 191-261.
[017] V. BUSH [1929], Operational Circuit Analysis, John Wiley, 392 pp. | Zbl 55.0838.01
[018] J. CAMPOS FERREIRA [1997], Introduction to the Theory of Distributions, Longman, 157 pp. | Zbl 0905.46001
[019] A. CAPELLI & GARBIERI [1894], Corso di analisi algebrica, Napoli. | Zbl 18.0048.01
[020] A. CAPELLI [1909], Istituzioni di analisi algebrica, Napoli, 953 pp. | Zbl 40.0136.04
[021] C. CARATHÉODORY [1963], Algebraic Theory of Measure and Integration, Chelsea, 378 pp. (Translation of 1956 German edition.)
[022] R. D. CARMICHAEL [1855], A Treatise on the Calculus of Operations: Designed to Facilitate the Processes of the Differential and Integral Calculus and the Calculus of Finite Differences, London, 170 pp.
[023] J. R. CARSON [1926], Circuit Theory and the Operational Calculus, McGraw-Hill, 197 pp. | Zbl 0050.41706
[024] A. L. CAUCHY [1821], Cours d'analyse. 1re Partie. Analyse algébrique, Oeuvres Ser. 2, Vol. 2, 476 pp.
[025] A. L. CAUCHY [1827], Exercices de mathématiques, Seconde année, Oeuvres, Ser. 2, Vol. 7.
[026] E. CESÀRO [1904], Elementares Lehrbuch der Algebraischen Analysis und der Infinitesimal Rechnung, Leipzig, 894 pp. Translation and revision of Corso di Analysi Algebrica, Torino, 1894, 499 pp.
[027] G. CHRYSTAL, G. [1904 ?], Algebra. An Elementary Textbook, 6th ed. Part II, 616 pp.
[028] R. M. COHN [1965], Difference Algebra, Interscience Press, 335 pp.
[029] J. L. B. COOPER [1952], Heaviside and the operational calculus, The Math. Gazette 36, 5-19. | Zbl 0046.11503
[030] J. P. DALTON [1954], Symbolic Operators, Johannesburg, 194 pp.
[031] H. T. DAVIS [1936], The Theory of Linear Operators, Bloomington, IN, 628 pp.
[032] M. A. B. DEAKIN [1981, 1982], The development of the Laplace transform, Arch. Hist. Exact Sci. I. Euler to Spitzer, 1737-1880, 25, 343-390; II. Poincaré to Doetsch, 1800--1937, 26, 351--381.
[033] A. De Morgan [1835], The Elements of Algebra. Preliminary to the Differential Calculus and Fit for the Higher Classes of Schools in Which the Principles of Arithmetic are Taught, London, 248 pp.
[034] A. De Morgan [1835] [1842], The Differential and Integral Calculus, London, 785 pp.
[035] A. De Morgan [1835] [1849], Trigonometry and Double Algebra, London, 167 pp.
[036] J. DIANNI [1977], The history of Jan Śniadecki's book 'The theory of the algebraic calculus applied to curved lines' in the light of his correspondence (in Polish), Kwart. Hist. Nauki i Tech. 22, No. 1, 59-71 [MR 58 (1979) #10079].
[037] G. DOETSCH [1937], Theorie und Anwendung der Laplace-Transformation, Springer, 436 pp. | Zbl 63.0368.01
[038] G. DOETSCH [1974], Introduction to the Theory and Application of the Laplace Transform, Springer, 326 pp. | Zbl 0278.44001
[039] L. EULER [1748], Introductio in Analysis Infinitorum, Vol. I, 392 pp.; Vol. II, 402 pp.
[040] L. EULER [1769], Institutiones calculi integralis, Vol. II, 676 pp. Opera Ser. I, Vol. XIII.
[041] J.-B. FOURIER [1822], Théorie Analytique de la Chaleur, Paris, 561 pp. | Zbl 15.0954.01
[042] J.-F. FRANÇAIS [1812-13], Mémoire tendant à démontrer la légitimité de la séparation des échelles des fonctions, Annales de Math. 3, 244-273.
[043] C. FRASER [1987], Joseph Louis Lagrange's algebraic version of the calculus, Historia Math. 14, 38-53. | Zbl 0617.01012
[044] J. GRABINER [1981], The Origin of Cauchy's Rigorous Calculus, M.I.T. Press, 252 pp. | Zbl 0517.01002
[045] J. GRABINER [1990], The Calculus as Algebra: J.-L. Lagrange, 1736-1813, Garland Publ. Co., Boston, 247 pp.
[046] J. GRABINER [1997], Was Newton's calculus a dead end? The continental influence of Maclaurin's Treatise of Fluxions, Amer. Math. Monthly 104, 393-410. | Zbl 0885.01010
[047] I. Grattan-Guinness [1994], Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge, Vol. I, 842 pp. | Zbl 0833.01001
[048] D. F. GREGORY [1841], Examples of the Processes of the Differential and Integral Calculus, Cambridge, 2nd ed., 1846, 529 pp.
[049] K. GÜRLEBECK and W. SPRÖSSIG [1997], Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley, 371 pp. | Zbl 0897.30023
[050] H. HANKEL [1867], Theorie der Complexen Zahlensystemes, Leipzig, 196 pp.
[051] O. D. HEAVISIDE [1893, 1899, 1912], Electromagnetic Theory, London, Vol. I, 466 pp.; Vol. II, 542 pp.; Vol. III, 519 pp. | Zbl 30.0801.03
[052] J. M. HOENE-WROŃSKI [1811], Introduction à la Philosophie des Mathématiques, Paris, 270 pp.
[053] J. M. HOENE-WROŃSKI [1811] [1812], Réfutation de la Théorie des Fonctions Analytiques de Lagrange, Paris, 136 pp.
[054] L. HÖRMANDER [1979], The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32, 359-443. | Zbl 0388.47032
[055] H. N. JAHNKE [1993], Algebraic analysis in Germany: 1780-% 1840, Historia Math. 20, 265-284. | Zbl 0784.01007
[056] J. H. JELLETT [1858], An Elementary Treatise on the Calculus of Variations, Dublin, 377 pp.
[057] I. KAPLANSKY [1957], Introduction to Differential Algebra, Hermann, 62 pp., 2nd ed., 1976.
[058] M. KASHIWARA & T. KAWAI, (eds.) [1988], Algebraic Analysis: Papers Dedicated to Professor Miko Sato on the Occasion of His Sixtieth Birthday, Academic Press, Vol. I, 472 pp.; Vol. II, pp. 473-950.
[059] M. KASHIWARA, T. KAWAI & T. KIMURA [1986], Foundations of Algebraic Analysis, Princeton University Press, 254 pp. | Zbl 0605.35001
[060] M. KASHIWARA & P. SCHAPIRA [1990], Sheaves on Manifolds, Springer, 512 pp. | Zbl 0709.18001
[061] E. R. KOLCHIN [1973], Differential Algebra and Algebraic Groups, Academic Press, 446 pp. | Zbl 0264.12102
[062] E. R. KOLCHIN [1985], Differential Algebraic Groups, Academic Press, 271 pp. | Zbl 0556.12006
[063] H. KÖNIG [1953], Neue Begrundung der Theorie der 'Distributionen' von L. Schwartz, Math. Nachr. 9, 129-148. | Zbl 0050.33802
[064] E. KOPPELMAN [1971-72], The calculus of operations and the rise of abstract algebra, Arch. Hist. Exact Sci. 8, 155-242. | Zbl 0232.01006
[065] S.-F. LACROIX [1797-1800], Traité du calcul différentiel et du calcul intégral, 3 Vol., 2nd ed., 1810-1814-1819, Courcier, Paris, Vol. I, 653 pp.; Vol. II, 819 pp.; Vol. III, 776 pp.
[066] S.-F. LACROIX [1802], Traité élémentaire du calcul différentiel et du calcul intégral, Duprat, Paris.
[067] S.-F. LACROIX [1816], An Elementary Treatise on the Differential and Integral Calculus, Cambridge. Translation of 1802 edition, 720 pp.
[068] J.-L. LAGRANGE [1772], Sur une nouvelle espèce de calcul relatif à la différentiation et à l'intégration des quantités variables, Nouvelles mémoires de l'acad. de Berlin, 185-221; Oeuvres, Vol. 3, pp. 439-476.
[069] J.-L. LAGRANGE [1797], Théorie des fonctions analytiques, Duprat, 276 pp.; 2nd ed., 1813, Oeuvres, Vol. 9, 427 pp.
[070] J.-L. LAGRANGE, [1799], Leçons sur le calcul des fonctions, 2nd ed., 1806, 501 pp.; Oeuvres, Vol. 10.
[071] J. LANDEN [1764], The Residual Analysis, a New Branch of the Algebraic Art, London, 218 pp.
[072] P. S. LAPLACE [1776], Mémoire sur l'inclinassion moyenne des orbites comètes sur la figure de la terre et sur les fonctions, Oeuvres, Vol. 8, 314-321.
[073] P. S. LAPLACE [1782], Mémoire sur les approximations des formules qui sont fonctions de très grands nombres, Mém. Acad. Roy. Sci. (Paris), 1-88, Oeuvres 10, 209-229.
[074] P. S. LAPLACE [1810-11], Mémoire sur les intégrales définies et leur application aux probabilités, Mém. Acad. Roy. Sci. (Paris) 9, 1-55; Oeuvres 10, 357-412.
[075] P. S. LAPLACE [1812], Théorie analytique des probabilités, 3rd ed., 1820, 774 pp. | Zbl 23.0020.04
[076] F. B. A. LEMBERT [1815], Handbuch der algebraischen Analysis, Köln.
[077] P. LÉVY [1926], Le calcul symbolique d'Heaviside, Bull. Sci. Math. 2, 174-192. Also published separately, Gauthier-Villars, 1926.
[078] J. LIOUVILLE [1832], Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, J. de L'École Polytech. 13, 1-69.
[079] T. P. G. LIVERMAN [1964], Generalized Functions and Direct Operational Methods, Prentice-Hall, Vol. I, Non-Analytic Functions in One Dimension, 338 pp. | Zbl 0119.10301
[080] J. LÜTZEN [1979], Heaviside's operational calculus and the attempts to rigorise it, Arch. Hist. Exact Sci. 21, 161-200. | Zbl 0419.01016
[081] C. MACLAURIN [1742], A Treatise of Fluxions in Two Books, Edinburgh, Vol. I, 416 pp.; Vol. II, 388 pp.
[082] P. MANSION [1898], Mélanges mathématiques, Gauthier-Villars, 160 pp. + 56 pp.
[083] J. G. MIKUSIŃSKI [1944], Hypernumbers, Kraków, 24 pp.
[084] J. G. MIKUSIŃSKI [1947], L'anneau algébrique et ses applications dans l'analyse fonctionnelle, Ann. Univ. Marie Curie-Skłodowska Sect. A 2, 1-48.
[085] J. G. MIKUSIŃSKI [1949], Sur les fondements du calcul opératoire, Studia Math. 2, 41-70.
[086] J. G. MIKUSIŃSKI [1953], Rachunek operatorów, PWN, 369 pp.
[087] I. NEWTON [1669], De Analysi per Aequationes Numero Terminorum Infinitas, published in 1711.
[088] L. NOVY [1973], Origins of Modern Algebra, Wolters-Noordhoff, 252 pp.
[089] G. PEACOCK [1830], A Treatise on Algebra, Cambridge; 2nd ed., Vol. I, Arithmetical Algebra, 1842, 399 pp.; Vol. II, On Symbolical Algebra and its Application to the Geometry of Position, 1846, 355 pp.
[090] G. PEACOCK [1834], Report on the recent progress and present state of certain branches of analysis, from: Report of the British Association for the Advancement of Science for 1833, London, 185-352.
[091] S. S. PETROVA [1987], Heaviside and the development of the symbolic calculus, Arch. Hist. Exact Sci. 37, 1-23. | Zbl 0624.01006
[092] J. PETZVAL [1853, 1859], Integration der linearen Differentialgleichungen mit Constanten Coefficienten, Braumiller, Vienna, Vol. I, 417 pp.; Vol. II, 687 pp.
[093] H. T. H. PIAGGIO [1952], An Elementary Treatise on Differential Equations and Their Applications, rev. ed., 256 pp. (1st ed., 1920). | Zbl 47.0418.03
[094] S. PINCHERLE [1887], Della trasformazione di Laplace e di alcune sue applicazioni, Mem. Accad. Bolgna 4, 125-143.
[095] H. POINCARÉ [1885], Sur les équations linéaires aux différentielles ordinaires et aux différences finies, Amer. J. Math. 7, 1-56; Oeuvres 1, 225-289. | Zbl 17.0290.01
[096] B. Van der Pol & H. BREMMER [1950], Operational Calculus Based on the Two-Sided Laplace Integral, Cambridge University Press, 2nd ed., 1955, 415 pp. | Zbl 0040.20403
[097] E. G. POOLE [1936], Introduction to the Theory of Linear Differential Equations, Clarendon Press, 202 pp. | Zbl 0014.05801
[098] D. PRZEWORSKA-ROLEWICZ [1988], Algebraic Analysis, D. Reidel & Polish Sci. Publ., 624 pp.
[099] D. PRZEWORSKA-ROLEWICZ [1988] [1997], Algebraic Analysis, M. Hazewinkel (ed.) Encyclopaedia of Mathematics. Supplement I, Kluwer, 42-43.
[100] J. Rey Pastor, P. Calleja & C. A. TREJO [1960], Análisis Matemático, Vol. I, Análisis algebraico - Teoria de equaciones, Cálcolo infinitesimal de una variable. 5th ed., 836 pp.; Vol. II, Cálcolo infinitesimal de varias variables - Applicaciones, 3rd ed., 1959, 624 pp.; Vol. III, Análisis funcional y applicaciones, 1959, 718 pp.
[101] B. RIEMANN [1953], Versuch einer allgemeinen Auffassung der Integration und Differentiation, Werke, 2nd ed., 353-366, Dover reprint.
[102] J. F. RITT [1932], Differential Equations From the Algebraic Standpoint, Amer. Math. Soc., 172 pp. | Zbl 58.0445.06
[103] J. F. RITT [1950], Differential Algebra, Amer. Math. Soc., 184 pp. | Zbl 0037.18402
[104] E. E. ROSINGER [1990], Nonlinear Partial Differential Equations. An Algebraic View of Generalized Solutions, North-Holland, 380 pp. | Zbl 0717.35001
[105] J. RYAN, (Ed.) [1996], Clifford Algebras in Analysis and Related Topics, CRC Press, 386 pp.
[106] X. SAINT-RAYMOND [1991], Elementary Introduction to the Theory of Pseudodifferential Operators, CRC Press, 108 pp. | Zbl 0847.47035
[107] R. SAUER [1958], Anfangswertprobleme bei Partiellen Differentialgleichungen, Springer, 2nd ed, 284 pp. | Zbl 0046.31902
[108] O. X. SCHLÖMILCH [1845], Handbuch der Algebraischen Analysis. | Zbl 20.0272.02
[109] L. SCHWARTZ [1950], Théorie des distributions, Hermann, Tome I, 2nd ed., 1957, 162 pp. | Zbl 0037.07301
[110] L. SCHWARTZ [1979], Comments on [26c], pp. 425-427 of Wiener [1979].
[111] J. SEBASTIÃO E SILVA [1954-55], Sur une construction axiomatique de la théorie des distributions, Rev. Fac. Ciências Lisbao 4, 79-186. | Zbl 0064.35802
[112] J. SEBASTIÃO E SILVA [1954-55] [1960], Sur la définition et la structure des distributions vectorielles, Portugaliae Math. 19, 1-80. | Zbl 0202.40303
[113] J. SEBASTIÃO E SILVA [1954-55] [1962], Sur l'axiomatique des distributions et ses possibles modèles, C. I. M. E., Rome, 15 pp.
[114] J. SEBASTIÃO E SILVA [1954-55] [1964], Integrals and orders of growth of distributions, in: Theory of Distributions. Proc. Intern. Instit. Lisbon, 1964, Lisbao, 327-390.
[115] F.-J. SERVOIS [1841-15], Essai sur un nouveau mode d'exposition des principes du calcul différentiel, Ann. Math. Pures Appl. 5, 93-170. Also published separately under the same title, Nismes, 1814.
[116] R. SIKORSKI [1954], A definition of the notion of distribution, Bull. Acad. Polon. Sci. Cl. III 2, 209-211. | Zbl 0055.10204
[117] R. SIKORSKI [1958], Funkcje Rzeczywiste (Real Functions), PWN, Vol. I, 534 pp.
[118] J. ŚNIADECKI [1783], Rachunku algebraicznego teoria przystosowana do linii krzywych (Theory of the Algebraic Calculus Applied to Curved Lines), 2 Vols. 504 pp.
[119] E. STEPHENS [1937], The Elementary Theory of Operational Mathematics, McGraw-Hill, 313 pp. | Zbl 0019.34801
[120] S. TAKAHASHI [1969], Analysis in Categories, Queen's University Papers, 131 pp.
[121] V. VOLTERRA [1913a], Leçons sur les équations intégrales et les équations intégro-différentielles, Gauthier-Villars, 164 pp. | Zbl 44.0409.03
[122] V. VOLTERRA [1913b], Leçons sur les fonctions de lignes, Gauthier-Villars, 164 pp.
[123] V. VOLTERRA and J. PÉRẼS [1924], Leçons sur la composition et les fonctions permutables, Gauthier-Villars, 183 pp. | Zbl 50.0290.04
[124] E. WARING [1776], Meditationes analyticae, Cambridge.
[125] H. WEYL [1927], Quantenmechanik und Gruppentheorie, Zeit. Physik 46, 1-46. | Zbl 53.0848.02
[126] H. WEYL [1928], Gruppentheorie und Quantenmechanik, Leipzig, 288 pp.; 2nd ed., 1931, 366 pp.; English translation of 2nd edition, 1932, 422 pp. | Zbl 54.0954.03
[127] N. WIENER [1926], The operational calculus, Math. Ann., 557-584. | Zbl 52.0416.01
[128] N. WIENER [1979], Collected Works With Commentaries, M.I.T. Press, Vol. II.
[129] R. WOODHOUSE [1803], The Principles of Analytical Calculation, Cambridge, 219 pp.
[130] R. WOODHOUSE [1809], A Treatise on Plane and Spherical Trigonometry, Cambridge; 3rd ed., 1819; 5th ed., 344 pp.