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Heaviside's operational calculus and the attempts to rigorise it

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Abstract

At the end of the 19th century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20th century several attempts were made to rigorise Heaviside's operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations (Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation (Lévy, Mikusiński). Also Schwartz's creation of the theory of distributions was very much inspired by problems in the operational calculus.

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Bibliography

  • Berg, E. J., 1929, Heaviside's Operational Calculus, New York 1929.

  • Boole, G., 1859, Treatise on Differential Equations, Cambridge 1859.

  • Bromwich, T. J., 1916, “Normal Coordinates in Dynamical Systems”, Proc. London Math. Soc. vol. 15, 1916, pp. 401–48.

    Google Scholar 

  • Bromwich, T. J., 1928, “Some Solutions of the Electromagnetic Equations, and of the Elastic Equations, with Applications to the Problem of Secondary Waves”, Proc. London Math. Soc. vol. 28, 1928, pp. 438–75.

    Google Scholar 

  • Bush, V., 1929, Operational Circuit Analysis, New York 1929.

  • Carslaw, H. S., 1928–29, “Operational Methods in Mathematical Physics”, Math. Gazette vol. 14, 1928–29, pp. 216–228.

    Google Scholar 

  • Carson, J. R., 1917, “On a General Expansion Theorem for Transient Oscillations of a Connected System”, Phys. Rev. ser. 2, vol. 10, 1917, pp. 217–25.

    Google Scholar 

  • Carson, J.R., 1919, “Theory of the Transient Oscillations of Electrical Networks and Transmission Lines”, Trans. Amer. Inst. Elec. Eng. vol. 38, 1919, pp. 345–427.

    Google Scholar 

  • Carson, J. R., 1922, “The Heaviside Operational Calculus”, Bell System Techn. Journ. vol. 1, 1922, pp. 43–55.

    Google Scholar 

  • Carson, J. R., 1926, Electric Circuit Theory and the Operational Calculus, New York 1926.

  • Casper, L., 1925, “Zur Formel von Heaviside für Einschaltvorgänge”, Arch. für Electrotechnik vol. 15, 1925, pp. 95–96.

    Google Scholar 

  • Cauchy, A. L., 1827, “Sur l'analogie des puissances et des différences”, “Addition à l'article précédent”, Exerc. de Math. vol. 2, 1827, pp. 159–92, 193–209, Oeuvres Complètes ser. 2, vol. 7, pp. 198–235, 236–54.

    Google Scholar 

  • Cohen, L., 1922, “The Heaviside Expansion Theorem”, Journ. Franklin Inst. vol. 194, 1922, pp. 765–70.

    Google Scholar 

  • Cooper, J.B.L., 1952, “Heaviside and the Operational Calculus”, Math. Gazette vol. 36, 1952, pp. 5–19.

    Google Scholar 

  • Dirac, G., 1926, “The Physical Interpretation of the Quantum Dynamics”, Proc. Roy. Soc. London ser. A, vol. 113, 1926, pp. 621–41.

    Google Scholar 

  • Doetsch, G., 1930, Besprechung von J. R. Carson, Elektrische Ausgleichsvorgänge und Operatorenrechnung, Jahresber. Deut. Math. Ver. vol. 39, 1930, pp. 105–109.

    Google Scholar 

  • Doetsch, G., 1937, Theorie und Anwendung der Laplace-Transformation, Berlin 1937.

  • Florin, H.B.J., 1934, Die Methoden der Hevisideschen Operatorenrechnung, Leiden 1934

  • Freudenthal, H., 1969, “Operatorenrechnung — von Heaviside bis Mikusiński”, Überblicke Mathematik vol. 2, 1969. Bibliograph. Inst. Mannheim. Ed. D. Laugwitz.

  • Gauster, W., 1930, “Die Lösung von Schwingungsaufgaben”, Arch. für Electrotechnik vol. 24, 1930, pp. 360–82.

    Google Scholar 

  • Heaviside, O., (EP) Electrical Papers, London 1892.

  • Heaviside, O., 1893, “On Operators in Physical Mathematics”, Proc. Roy. Soc. London vol. 52, 1893, pp. 504–29.

    Google Scholar 

  • Heaviside, O., (EMT) Electromagnetic Theory, vol. I London 1893, vol. II London 1899, vol. III London 1912.

  • Leffreys, H., 1927, Operational Methods in Mathematical Physics, Cambridge 1927.

  • Kirchhoff, G., 1891, Vorlesungen über Mathematische Physik II, Leipzig 1891.

  • Koppelman, E., 1971–72, “The Calculus of Operations and the Rise of Abstract Algebra”, Arch. Hist. Ex. Sci. vol. 8, 1971–72, pp. 155–242.

    Google Scholar 

  • Kuhn, T. S., 1962, The Structure of Scientific Revolutions, Chicago 1962.

  • Laplace, P. S. de, 1812, Théorie analytique des probabilités, Paris 1812.

  • Lévy, P., 1926, “Le calcul symbolique de Heaviside”, Bull. Sci. Math. (2) vol. 50, 1926, p. 174–192.

    Google Scholar 

  • Locher, L., 1934, “Zur Auflösung eines Systems von linearen gewöhnlichen Differentialgleichungen mit konstanten Koeffizienten”, Comment. Math. Helv. vol. 7, 1934, pp. 47–62.

    Google Scholar 

  • March, H. W., 1927, “The Heaviside Operational Calculus”, Bull. Amer. Math. Soc. vol. 33, 1927, pp. 311–18.

    Google Scholar 

  • Mikusiński, J., 1950, “Sur les fondements du calcul opératoire”, Studia Mathematica vol. 11, 1950, pp. 41–70.

    Google Scholar 

  • Mikusiński, J., 1959, Operational Calculus, Warszawa 1959.

  • Neumann, J. von, 1927, “Mathematische Begründung der Quantenmechanik”, Nachr. Ges. Wiss. Göttingen Math. Phys. Kl., 1927, pp. 1–57.

  • Pincherle, S., 1904–16, “Funktionaloperationen und -Gleichungen”, Enc. Math. Wiss., 2. Band, 1. Teil 2, 1904–1916, pp. 761–817.

    Google Scholar 

  • Pol, B. Van der, 1929, “A Simple Proof and an Extension of Heaviside's Operational Calculus for Invariable Systems”, Phil. Mag. vol. 7, 1929, pp. 1153–62.

    Google Scholar 

  • Pol, B. van der, & H. Bremmer, 1950, Operational Calculus Based on the Two-Sided Laplace-Integral, New York 1950.

  • Ross, B., 1977, “The Development of Fractional Calculus 1695–1900”, Hist. Math. vol. 4, 1977, pp. 75–89.

    Google Scholar 

  • Schwartz, L., 1945, “Généralisation de la notation de fonction, de dérivation, de transformation de Fourier, et applications mathématiques et physiques”, Ann. Univ. Grenoble vol. 21, 1945, pp. 57–74.

    Google Scholar 

  • Schwartz, L., 1947, “Théorie des distributions et transformation de Fourier”, Colloques Internationaux. Analyse Harmonique, 1947, pp. 1–8.

  • Servois, F.J., 1814–15, “Réflexions sur les divers systèmes d'exposition des principes du calcul différentiel, et en particulier, sur la doctrine des infiniment petits”, Ann. Math. Pures Appl. vol. 5, 1814–15, pp. 141–70.

    Google Scholar 

  • Smith, J. J., 1925, “An Analogy Between Pure Mathematics and the Operational Mathematics of Heaviside by Means of the Theory of H-Functions”, Journ. Franklin Inst. vol. 200, 1925, pp. 519–34, 635–72, 775–814.

    Google Scholar 

  • Titchmarsh, E. C., 1926, “The Zeros of Certain Integral Functions”, Proc. London Math. Soc. vol. 25, 1926, pp. 283–302.

    Google Scholar 

  • Vallarta, M. S., 1926, “Heaviside's Proof of His Expansion Theorem”, Trans. Am. Inst. Elec. Eng. vol. 65, 1925, pp. 429–34.

    Google Scholar 

  • Wagner, K. W., 1915–16, “Über eine Formel von Heaviside zur Berechnung von Einschaltvorgängen”, Arch. Electrotechnik vol. 4, 1915–16, pp. 159–93.

    Google Scholar 

  • Wagner, K. W., 1925, “Der Satz von der wechselseitigen Energie”, Elektr. Nachricht-Technik, vol. 2, 1925, pp. 376–392.

    Google Scholar 

  • Whittaker, E. T., 1928–29, “Oliver Heaviside”, Bull. Calcutta Math. Soc., vol. 20, 1928–29, pp. 199–220.

    Google Scholar 

  • Wiener, N., 1926, “The operational calculus”, Math. Ann., vol. 95, 1926, pp. 557–84.

    Google Scholar 

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  1. History of Science Department, University of Aarhus, Denmark

    Jesper Lützen

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Communicated by H. Freudenthal

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Lützen, J. Heaviside's operational calculus and the attempts to rigorise it. Arch. Hist. Exact Sci. 21, 161–200 (1979). https://doi.org/10.1007/BF00330405

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