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On the early history of Bessel functions

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Bibliography

  • Bernoulli, D. [1] “Observationes de seriebus recurrentibus,” Comment. Acad. Sci. Petropolis Vol III, 1728 (1732), 85–100.

    Google Scholar 

  • Bernoulli, D. [2] “Theoremata de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae.” Ibid. Vol 6, 1732-33, (1738), 108–122. (See Cannon & Dostrovsky [1; Appendix] for a translation into English.)

    Google Scholar 

  • Bernoulli, D. [3] “Demonstrationes theorematum suorum de oscillationibus corporum filo flexili connexorum et catenae verticaliter suspensae,” Ibid Vol 7, 1734–35, (1740), 162–173. (See Cannon & Dostrovsky [1; Appendix] for a translation into English.)

    Google Scholar 

  • Bessel, F. W. [l] “Ueber das Dollons'sche Mittagsfernrohr und den Cary'schen Kreis,” Abhandlungen, Leipzig 1875–76, II, 19–32 Königsberger Bericht, Abh. (1815), III–XXXI.

  • Bessel, F. W. [2] “Analytische Auflösung der Kepler'schen Aufgabe,” Abhandlungen der Berliner Akademie (1816–17), publ. 1819, 49–55, Repr. Abhandlungen, I, (1875), 17–20.

  • Bessel, F. W. [3] “Untersuchung des Teils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht,” Abhandlungen der Berliner Akademie (1824), publ. 1826, 1–52. Repr. Abhandlungen I, (1875), 84–109.

  • Bôcher, M. [1] “Introduction to the Theory of Fourier Series,” Annals of Mathematics, Ser. 2 Vol 7 (1906), 81–152.

    Google Scholar 

  • Burkhardt, H. , & Esclangon, E. [1] “Interpolation Trigonometrique,” Encycl. des Sciences Mathématique, (II, 27) (1912).

  • Burkhardt, H. [1] “de Entwicklungen nach oscillierenden Functionen und Integration der Differentialgleichungen der mathematischen Physik,” Jahresber. der Deutsch. Mathematiker Vereinigung, Bd 10, Heft 2 (1901-1908) 1800 pp.

  • Burkhardt, H. [2] “Trigonometrische Reihen und Integrale", Encykl. der Math. Wissenschaften, Bd II, Teil 1, Hälfte II, Heft 12 (1916).

  • Cannon, J. T., & Dostrovsky, S. The Evolution of Dynamics: Vibration Theory from 1687 to 1742, New York, 1981.

  • Carlini, F. [1] “Richerche sulla convergensa della serie che serva alla soluzione del problema di Kepler” Milan, 1817. Translated into German by C. G. J. Jacobi in Gesammelte Werke Vol VII, 189–245, and originally published in 1850.

  • Davis, H. [1] Fourier Series and Orthogonal Functions, Boston, 1963.

  • Davis, P. J., & Hersh, R. [1] The Mathematical Experience, Boston, 1982.

  • Euler, L. [1] “De oscillationibus fili flexilis quotcunque ponduscilis onusti, E49, 1736, Opera Omnia, Ser 2, Vol 10, 38–49.

    Google Scholar 

  • Euler, L. [2] “Sur la force des colonnes,” E-238, Memoires de l'Academie de Berlin T.XIII 1757, (1759), 252–282, Repr. Opera Omnia, Ser 2 Vol 17, 89–118.

  • Euler, L. [3] Institutionum Calculi Integralis, Vol 2, Petropolis, 1769 Repr. in Opera Omnia, Ser 1, Vol 12.

  • Euler, L. [4] “De motu vibratorio tympanorum” E302, Nov. Comm. Acad. Petropolitanae, 1778 (1780). Repr. Opera Omnia, Ser 2, Vol 17, 252–265.

  • Euler, L. [5] “Determinatio onerum quae columnae gestare valent,” E508, Acta Acad. Sci. Petropolis, Vol 2, 1778, (1780), 121–145. Repr. Opera Omnia, Ser 2 Vol 17, 232–251.

    Google Scholar 

  • Euler, L. [6] “Examen insignis paradoxi in theoria columnarum occurentis”, E 509 Acta Acad. Petrop., 1778 [1780], Vol 1, 146–162. Repr. Opera Omnia Ser 2, Vol. 17, 252–265.

    Google Scholar 

  • Euler, L. [7] “De oscillationibus minimis funis libere suspensi”, E576, (1781), Opera Omnia, Ser 2, Vol 11, 307–323.

    Google Scholar 

  • Euler, L. [8] “De perturbatione motus chordarum ab earum pondere oriunda,” E577, (1781), Opera Omnia Ser 2, Vol 11, 324–334.

    Google Scholar 

  • Fourier, J. [1] Théorie analytique de la chaleur, Paris 1822. An English translation by A. Freeman with notes, appeared in 1876. A reprint of this was published in New York in 1955.

  • Grattan-Guiness, I. [1] The Development of the Foundations of Mathematical Analysis from Euler to Riemann, Cambridge, (Mass), 1970.

  • Grattan-Guiness, I. [2] Joseph Fourier 1768–1830, Cambridge (Mass), 1972.

  • Greenhill, A. G. [1] “Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow,” Proceedings of the Cambridge Philosophical Society, Vol IV (1883), 65–73.

    Google Scholar 

  • Kepler, J. [1] Astronomia Nova..., 1609. Translated into English under the title New Astronomy by W. H. Donahue, Cambridge, 1992.

  • Kline, M. [1] Mathematical Thought from Ancient to Modern Times, New York, 1972.

  • Laplace, P. S. [1] Traité de Mecanique Céleste, 5 vols. Paris 1825. Repr. New York, 1969.

  • Lagrange, J. L. L. [1] “Sur le problème de Kepler,” Hist. de l'Acad. R. des Sciences de Berlin, Vol XXV, 1770 (1771), 204–233. Repr in Oeuvres, Vol III, 113–126.

    Google Scholar 

  • Langer, R. E. [1] “Fourier's Series, The Genesis and Evolution of a Theory”, American Mathematical Monthly, Vol 54, No. 7, Part II (1947).

  • Leibniz, G. W. [1] Mathematische Schriften, Halle, 1855, Repr. Hildesheim 1962. (The first reference in Watson [1; 744] to Daniel Bernoulli should be to James Bernoulli.)

  • Maggi, G. A. [1] “Sulla storia delle funzioni cilindriche,” Atti della R. Accademia del Lincei, (Transunti), Anno 277, Ser 3, Vol IV, (1880), 259–263.

    Google Scholar 

  • Miller, W. [1], Lie Theory and Special Functions, New York, 1968.

  • Poisson, S. D. [1] “Second mémoire sur la distribution de la chaleur dans les corps solides,” Journal de l'Ecole Polytechniques T. XII, Cahier 19, (1823), 249–403.

    Google Scholar 

  • Schlömilch, O. [1] “Üeber die Bessel'schen Function,” Zeitschrift für Mathematik und Physik, II (1857), 137–165.

    Google Scholar 

  • Sommerfeld, A. [1] Partial Differential Equations in Physics, New York, 1949.

  • Szegö, G. [1] Orthogonal polynomials, New York, 1939.

  • Taylor, B. [1] “De motu nervi tensi”, Phil. Trans. of the R. S. of London Vol. 28 (1713), 28–32.

    Google Scholar 

  • Truesdell, C. [1] “An essay toward a Unified Theory of Special Functions based upon the functional equation \(\frac{\partial }{{\partial z}}F\left( {z,\alpha } \right) = F\left( {z,\alpha + 1} \right)\),” Annals of Mathematics Studies, No. 18, Princeton, 1948.

  • Truesdell, C. [2], “The Rational Mechanics of flexible or Elastic Bodies 1638–1788”. Euler, L. Opera Omnia, Ser 2, Vol XI, Sect. II, Zürich, 1960.

  • Wagner, C. [1] “Beiträge zur Entwicklung der Bessel'schen Funktion I. Art,” Mitteilungen der naturforschenden Gesellschaft in Bern, 1894, 204–266.

  • Watson, G. N. [1] A Treatise on the Theory of Bessel Functions, 2nd edition Cambridge, 1944.

  • Whittaker, E. T., & Watson, G. N. [1] A Course of Modern Analysis, Cambridge, 4th edition, 1945.

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Communicated by Curtis Wilson

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Dutka, J. On the early history of Bessel functions. Arch. Hist. Exact Sci. 49, 105–134 (1995). https://doi.org/10.1007/BF00376544

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