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Archive for History of Exact Sciences

, Volume 26, Issue 4, pp 351–381 | Cite as

The development of the Laplace Transform, 1737–1937 II. Poincaré to Doetsch, 1880–1937

  • Michael A. B. Deakin
Article

Abstract

An earlier paper, to which this is a sequel, traced the history of the Laplace Transform up to 1880. In that year Poincaré reinvented the transform, but did so in a more powerful context, that of properly conceived complex analysis. Rapid developments followed, culminating in Doetsch' work in which the transform took its modern shape.

Keywords

Rapid Development Complex Analysis Early Paper Laplace Transform Powerful Context 
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References

  1. 1.
    Abel, N. H., Sur les fonctions génératrices et leurs déterminantes. In Oeuvres Complètes de N. H. Abel (Ed. B. Holmboe) Tome Second (Christiania: Grøndahl, 1839), 77–88. Also in Oeuvres Complètes de Niels Henrik Abel (Ed. L. Sylow & S. Lie) Tome Second (Christiania: Grøndahl, 1881), 67–81.Google Scholar
  2. 2.
    Amaldi, U., Sulla trasformazione di Laplace. Atti R. Accad. Lincei, 5 ser., Rend. Classe sci. fis. mat. nat. 7 (1898), 117–124.Google Scholar
  3. 3.
    Bateman, H., The solution of linear differential equations by means of definite integrals. Trans. Camb. Phil. Soc. 21 (1909), 171–196.Google Scholar
  4. 4.
    Bateman, H., On the inversion of a definite integral. Proc. Lond. Math. Soc., 2 ser. 4 (1906, 1907), 461–498.Google Scholar
  5. 5.
    Bateman, H., Report on the history and present state of the theory of integral equations. Rep. 80th Meet. Br. Ass. Adv. Sci. (1910, 1911), 345–424.Google Scholar
  6. 6.
    Bateman, H., The solution of a system of differential equations occurring in the theory of radio-active transformations. Proc. Camb. Phil. Soc. 15 (1910), 423–427.Google Scholar
  7. 7.
    Bateman, H., The control of an elastic fluid. Bull. Amer. Math. Soc. 51 (1945), 601–646.Google Scholar
  8. 8.
    Bernstein, F., Die Integralgleichung der elliptischen Thetanullfunktion. Zweite Note: Allgemeine Lösung. K. Akad. Wetens. Amsterdam Vers. (1920), 759–765.Google Scholar
  9. 9.
    Bernstein, F., Ueber die numerische Ermittlung verborgener Periodizitäten. Z. ang. Math. Mech. 7 (1927), 441–444.Google Scholar
  10. 10.
    Bernstein, F., & G. Doetsch, Die Integralgleichung der elliptischen Thetanullfunktion. Dritte note. Dritte Herleitung durch den verallgemeinerten Volterraprozess und weitere Beispiele (Mittag-Leffler Funktion und Beltramische Integralgleichung). Nach. K. Gessell. Wissens. Göttingen. Math.-phys. Klasse (1921, 1922), 32–46.Google Scholar
  11. 11.
    Bernstein, F., & G. Doetsch, Die Integralgleichung der elliptischen Thetanullfunktion. Vierte Note. Integrale Additionstheoreme. Das Additionstheorem von Cesàro und das der Funktion U(c/t). Vierte Herleitung der fundamentalen Integralgleichung. Nach. K. Gessell. Wissens. Göttingen Math.-phys. Klasse (1921, 1922), 47–52.Google Scholar
  12. 12.
    Bernstein, F., & G. Doetsch, Über die Integralgleichung der elliptischen Thetanullfunktion. Jahresb. Deutsche. Math. Ver. 31 (1922), 148–153.Google Scholar
  13. 13.
    Bernstein, F., & G. Doetsch, Probleme aus der Theorie der Wärmeleitung. I. Mitteilung. Eine neue Methode zur Integration partieller Differentialgleichungen. Der lineare Wärmeleiter mit verschwindender Anfangstemperatur. Math. Z. 20 (1924), 285–292.Google Scholar
  14. 14.
    Bernstein, F., & G. Doetsch, Probleme aus der Theorie der Wärmeleitung. IV. Mitteilung. Die räumliche Fortsetzung des Temperaturablaufs (Bolometerproblem). Math. Z. 26 (1927), 89–98.Google Scholar
  15. 15.
    Birkhoff, G. D., Singular points of ordinary linear differential equations. Trans. Amer. Math. Soc. 10 (1908, 1909), 436–470.Google Scholar
  16. 16.
    Birkhoff, G. D., On a simple type of irregular singular point. Trans. Amer. Math. Soc. 14 (1911, 1913), 462–476.Google Scholar
  17. 17.
    Boole, G., A Treatise on Differential Equations (Cambridge: Macmillan, 1859; 2nd Edn. ibid. 1865; 2nd Edn. reprinted as 5th Edn.: New York: Chelsea, n.d. [1959?]).Google Scholar
  18. 18.
    Borel, É., Sur les séries de Taylor. Acta Math. 20 (1897), 243–247.Google Scholar
  19. 19.
    Borel, É., Mémoire sur les séries divergentes. Ann. Éc. Norm. Sup. 16 (1899), 1–136 (esp. pp. 50-72).Google Scholar
  20. 20.
    Borel, É., Leçons sur les séries divergentes (Paris: Gauthier-Villars, 1901) (esp. pp. 97–115).Google Scholar
  21. 21.
    Borůvka, O., Article on Lerch in Dictionary of Scientific Biography, Vol. VIII. (Ed. C. C. Gillespie et al.) (New York: Scribner 1973), 253.Google Scholar
  22. 22.
    Bromwich, J. T. I'A., An Introduction to the Theory of Infinite Series (London: Macmillan, 1908); esp. pp. 276–283.Google Scholar
  23. 23.
    Bromwich, J. T. I'A., Normal coordinates in dynamical systems. Proc. Lond. Math. Soc. 2 ser. 15 (1914, 1916), 401–448.Google Scholar
  24. 24.
    Bromwich, J. T. I'A., Examples of operational methods in mathematical physics. Phil. Mag. 37 (1919), 407–419.Google Scholar
  25. 25.
    Bromwich, J. T. I'A., Some solutions of the electromagnetic equations, and of the elastic equations, with applications to the problem of secondary waves. Proc. Lond. Math. Soc. 28 (1927, 1928), 438–475.Google Scholar
  26. 26.
    Busbridge, I. W., On general transforms with kernels of the Fourier type. J. Lond. Math. Soc. 9 (1934), 179–187.Google Scholar
  27. 27.
    Carson, J. R., On a general expansion theorem for the transient oscillations of a connected system. Phys. Rev. 10 (1917), 217–225.Google Scholar
  28. 28.
    Carson, J. R., The Heaviside operational calculus. Bell. Syst. Tech. J. 1 (1922), 43–55.Google Scholar
  29. 29.
    Carson, J. R., Electric Circuit Theory and the Operational Calculus (New York: 1926; 2nd Edn. 1953).Google Scholar
  30. 30.
    Cooper, J. L. B., Heaviside and the operational calculus. Math. Gaz. 36 (1952), 5–18.Google Scholar
  31. 31.
    Deakin, M. A. B., Euler's version of the Laplace Transform. Am. Math. Monthly 87 (1980), 264–269.Google Scholar
  32. 32.
    Deakin, M. A. B., The development of the Laplace Transform, 1737–1937 I. Euler to Spitzer, 1737–1880. Arch. Hist. Ex. Sci. 25 (1981), 343–390.Google Scholar
  33. 33.
    Deakin, M. A. B., Motivating the Laplace Trapsform. Int. J. Math. Ed. Sci. Tech. 12 (1981), 415–418.Google Scholar
  34. 34.
    Doetsch, [G.], Review of Elektrische Ausgleichsvorgänge und Operatorenrechnung (German translation of Ref. [29]) by J. R. Carson. Jahresb. Deutsche. Math. Ver. 39 (1930), Literarisches 105–109.Google Scholar
  35. 35.
    Doetsch, G., Überblick über Gegenstand und Methode der Funktionanalysis. Jahresb. Deutsche. Math. Ver. 36 (1927), 1–30.Google Scholar
  36. 36.
    Doetsch, G., Theorie und Anwendung der Laplace-Transformation (Berlin: Springer 1937; 2nd Edn. New York: Dover, 1943). This book was extensively revised and expanded as Handbuch der Laplace-Transformation (Basel: Birkhäuser; Bd. I, 1950; Bd. II, 1955; Bd. III, 1956). In the revision, the reference list is greatly extended, but mainly by the inclusion of much later material. Apart from the notice given to two minor papers by Schlömilch, all relevant differences between the two versions are noted in the text.Google Scholar
  37. 37.
    Erdélyi, A., Harry Bateman 1882–1946. Obit. Not. Fell. Roy. Soc. 15 (1947), 590–618.Google Scholar
  38. 38.
    Erdélyi, A. (Editor), Tables of Integral Transforms (Based in part, on notes left by Harry Bateman), Vols I, II (New York: McGraw-Hill, 1954).Google Scholar
  39. 39.
    Fock, V., Über eine Klasse von Integralgleichungen. Math. Z. 21 (1924), 161–173.Google Scholar
  40. 40.
    Goursat, É., Cours d'analyse mathématique, Tome II (Paris: Gauthier-Villars, 1905).Google Scholar
  41. 41.
    Hardy, G. H., Notes on some points in the integral calculus. LVIII. On Hilbert transforms. Mess. Math. 54 (1924), 20–27.Google Scholar
  42. 42.
    Heaviside, O., Electromagnetic Theory, Vol. III (London: ‘The Electrician’, 1912; Reprinted, as 3rd Edn., New York: Chelsea, 1971).Google Scholar
  43. 43.
    Hermite, C. (rapporteur), Grand prix des sciences mathématiques. C. R. Acad. Sci. 92 (1881), 551–554. Also in Oeuvres de Henri Poincaré 2 (Ed. G. Darboux) (Paris: Gauthier-Villars, 1916), 71–74.Google Scholar
  44. 44.
    Horn, J., Über das Verhalten der Integrale von Differentialgleichungen bei der Annäherung der Veranderlichen an eine Unbestimmheitsstelle (Erster Teil). J. reine ang. Math. (Crelle) 118 (1897), 257–274.Google Scholar
  45. 45.
    Horn, J., Sur les intégrales irrégulières des équations différentielles linéaires. C. R. Acad. Sci. (Paris) 126 (1898), 205–208.Google Scholar
  46. 46.
    Horn, J., Verallgemeinerte Laplacesche Integrale als Lösungen linearer und nichtlinearer Differentialgleichungen. Jahresb. Deutsche. Math. Ver. 25 (1917), 301–325.Google Scholar
  47. 47.
    Horn, J., Laplacesche Integrale, Binomialkoeffizientenreihen und Gammaquotientenreihen in der Theorie der linearen Differentialgleichungen. Math. Z. 21 (1924), 85–95.Google Scholar
  48. 48.
    Ince, E. L., Ordinary Differential Equations (London: Longmans, 1927; Reprinted New York: Dover 1965).Google Scholar
  49. 49.
    Jeffreys, H., Operational Methods in Mathematical Physics (Cambridge University Press, 1927).Google Scholar
  50. 50.
    Jeffreys, H., Heaviside's pure mathematics. In The Heaviside Centenary Volume (London: Inst. Elec. Eng., 1950), pp. 90–92. Reprinted in Collected Papers of Sir Harold Jeffreys on Geophysics and Other Sciences, Vol. 6 (London: Gordon & Breach, 1977), pp. 123–128.Google Scholar
  51. 51.
    Jeffreys, H., & D. P. Dalzell, On the Heaviside operational calculus. Proc. Camb. Phil. Soc. 36 (1940), 267–282. Reprinted in Collected Papers of Sir Harold Jeffreys on Geophysics and Other Sciences, Vol. 6 (London: Gordon & Breach, 1977), pp. 101–116.Google Scholar
  52. 52.
    Jordan, C., Cours d'Analyse 3 (Paris: Gauthier-Villars, 1887). Pagination varies slightly in later editions.Google Scholar
  53. 53.
    Josephs, H. J., The Heaviside papers found at Paignton. In Electromagnetic Theory, Vol. 3 (3rd Edn.) (New York: Chelsea, 1971), 643–666.Google Scholar
  54. 54.
    Laplace, P. S., Mémoire sur les approximations des formules qui sont fonctions de très grands nombres. Mem. Acad. roy. Sci. (Paris) (1782, 1785), 1–88. Also in Oeuvres Complètes de Laplace (Paris: Gauthier-Villars, 1878–1912) 10, 209–291.Google Scholar
  55. 55.
    Lerch, M., Základové theorie Malmsténovskych rad. Rozpravy České Akad. C. Frant. Josefa 2 ser. 1 (1892), Číslo 27 (70 pp.).Google Scholar
  56. 56.
    Lerch, M., O hlavní větě theorie funkcí vytvořujících. Rozpravy České Akad. C. Frant. Josefa 2 ser. 1 (1892), Číslo 33 (7 pp.).Google Scholar
  57. 57.
    Lerch, M., Z počtu integrálního. Rozpravy České Akad. C. Frant. Josefa 2 ser. 2, Číslo 9 (40 pp.).Google Scholar
  58. 58.
    Lerch, M., Sur un point de la théorie des fonctions génératrices d'Abel. Acta Math. 27 (1903), 339–351.Google Scholar
  59. 59.
    Lévy, P., Le calcul symbolique d'Heaviside. Bull. Sci. Math. 2 ser. 50 (1926), 174–192. Also published under the same title as a pamphlet (Paris: Gauthier-Villars, 1926). [Omitted from Oeuvres de Paul Lévy 1 (Ed. D. Dugué) (Paris: Gauthier-Villars, 1963), Section III: Calcul Symbolique.]Google Scholar
  60. 60.
    Lützen, J., Heaviside's operational calculus and the attempts to rigorise it. Arch. Hist. Ex. Sci. 21 (1979), 161–200.Google Scholar
  61. 61.
    Macdonald, H. M., Some applications of Fourier's theorem. Proc. Lond. Math. Soc. 35 (1902, 1903), 428–443.Google Scholar
  62. 62.
    March, H. W., The Heaviside operational calculus. Bull. Amer. Math. Soc. 33 (1927), 311–318.Google Scholar
  63. 63.
    Martis in Biddau, S., Studio della trasformazione di Laplace e della sua inversa dal punto di vista dei funzionali analitici. Rend. Circ. Mat. Palermo 57 (1933), 1–70.Google Scholar
  64. 64.
    Mellin, Hj., Om definita integraler, hvilka för obergrändsat växande värden af vissa heltaliga parameterar hafva till gränser hypergeometriska functioner af särskila ordninger. Acta. Soc. Sc. Fenn. 20 (1895), No. 7 (39 pp.).Google Scholar
  65. 65.
    Mellin, Hj., Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma- und der hypergeometrischen Functionen. Acta Soc. Sc. Fenn. 21 (1896), No. 1 (115 pp.).Google Scholar
  66. 66.
    Mellin, Hj., Über gewisse durch bestimmte Integrale vermittelte Beziehung zwischen linearen Differentialgleichungen mit rationalen Coefficienten. Acta Soc. Sc. Fenn. 21 (1896), No. 6 (57 pp.).Google Scholar
  67. 67.
    Nathan, H., Article on Bernstein in Dictionary of Scientific Biography Vol. II (Ed. C. C. Gillespie et al.) (New York: Scribner, 1970), 58–59.Google Scholar
  68. 68.
    Paley, R. E. A. C., & N. Wiener, Fourier Transforms in the Complex Domain (Providence: Amer. Math. Soc., 1934).Google Scholar
  69. 69.
    Picard, É., Sur la transformation de Laplace et les équations linéaires aux dérivées partielles. C. R. Acad. Sci. (Paris) 107 (1888), 594–597.Google Scholar
  70. 70.
    Picard, É., Sur une classe d'équations aux dérivées partielles du second ordre. Rend. Circ. Mat. Palermo 5 (1891), 308–318.Google Scholar
  71. 71.
    Picard, É., Traité d'Analyse, Tome III (Paris: Gauthier-Villars, 1908).Google Scholar
  72. 72.
    Picard, É., Leçons sur quelques types simples d'équations aux dérivées partielles avec des applications à la physique mathématique (Fascicule 1 of Cahiers scientifiques, ed. G. Julia) (Paris: Gauthier-Villars, 1927; reprinted 1950).Google Scholar
  73. 73.
    Pincherle, S., Sopra una trasformazione delle equazioni differenziali lineari alle differenze, e viceversa. Rend. R. Ist. Lomb. 2 ser. 19 (1886), 559–562.Google Scholar
  74. 74.
    Pincherle, S., Della trasformazione di Laplace e di alcune sue applicazioni. Mem. Accad. Bologna, 4 ser. 8 (1887), 125–143.Google Scholar
  75. 75.
    Pincherle, S., Die Transformation von Laplace. Section 16 of Funktionaloperationen und -gleichungen in Enc. Math. Wiss. II, 1, 2 (Ed. H. Burkhardt & W. Wirtinger) (Leipzig: Teubner, 1905) [IIAII], 781–784.Google Scholar
  76. 76.
    Pincherle, S., Sulle operazioni funzionali lineari. Proc. Int. Math. Cong. Toronto 1 (1924, 1928), 129–137.Google Scholar
  77. 77.
    Pincherle, S., & U. Amaldi, Le Operazioni Distributive e le loro Applicazioni all' Analisi (Bologna: Zanichelli, 1901).Google Scholar
  78. 78.
    Plancherel, M., Contribution à l'étude de la représentation d'une fonction arbitraire par les intégrales définies. Rend. Circ. Mat. Palermo 30 (1910), 289–335.Google Scholar
  79. 79.
    Plancherel, M., Sur la convergence et sur la sommation par les moyennes de Cesàro de \(\mathop {\lim }\limits_{z \to \infty } {\text{ }}\int\limits_a^z {f(} x){\text{ cos }}xy dx\). Math. Ann. 76 (1915), 315–326.Google Scholar
  80. 80.
    Poincaré, H., Sur les fonctions fuchsiennes (Extrait d'un mémoire inédit). Acta Math. 39 (1923), 58–93. Also in Oeuvres d'Henri Poincaré (Paris: Gauthier-Villars, 1928- ) (hereinafter referred to as O.H.P.) Vol. 1, 336–373.Google Scholar
  81. 81.
    Poincaré, H., Notice sur les travaux scientifiques de M. Henri Poincaré. (Paris: Gauthier-Villars, 1884).Google Scholar
  82. 82.
    Poincaré, H., Sur les équations linéaires aux differentielles ordinaires et aux différences finies. Am. J. Math. 7 (1885), 1–56. O.H.P. 1, 225–289.Google Scholar
  83. 83.
    Poincaré, H., Sur les intégrales irrégulières des équations linéaires. Acta Math. 8 (1886), 295–344. O.H.P. 1, 290–332.Google Scholar
  84. 84.
    Poincaré, H., Remarques sur les intégrales irrégulières des équations linéaires (Réponse à M. Thomé). Acta Math. 10 (1887), 310–312. O.H.P. 1, 333–335.Google Scholar
  85. 85.
    Poincaré, H., Sur la théorie des quanta. C. R. Acad. Sci. (Paris) 153 (1911), 1103–1108. O.H.P. 9, 620–625.Google Scholar
  86. 86.
    Poincaré, H., Sur la théorie des quanta. J. Phys. théor. appl. 5 ser. 2 (1912), 5–34. O.H.P. 9, 626–653.Google Scholar
  87. 87.
    Poincaré, H., Analyse des travaux scientifiques de Henri Poincaré. Acta Math. 38 (1921), 4–125. This reference updates References [81]. It also is to be found (in sections) in O.H.P. The most relevant is that of Vol. 1, pp. I-XXXV.Google Scholar
  88. 88.
    Poincaré, H. (& E. Picard rapporteurs), Grand prix des sciences mathématiques. C. R. Acad. Sci. (Paris) 127 (1898), 1061–1065. (Note: The title page and the index (p. 1298) list the authors of this report as Picard & Poincaré. However, elsewhere (p. 1261), the index lists Poincaré as the rapporteur.)Google Scholar
  89. 89.
    Poisson, S. D., Mémoire sur la distribution de la chaleur dans les corps solides. J. Éc. Roy. Polytech 12 (19 cah.) (1923), 1–162 (esp. 23–34).Google Scholar
  90. 90.
    van der Pol, B., A simple proof and an extension of Heaviside's operational calculus for invariable systems. Phil. Mag. 7 ser. 7 (1929), 1153–1162.Google Scholar
  91. 91.
    van der Pol, B., On the operational solution of linear differential equations and an investigation of the properties of these equations. Phil. Mag. 7 ser. 8 (1929), 861–898.Google Scholar
  92. 92.
    van der Pol, B., A new theorem on electrical networks. Physica 4 (1937), 585–589.Google Scholar
  93. 93.
    van der Pol, B., Application of the operational or symbolic calculus to the theory of prime numbers. Phil. Mag. 7 ser. 26 (1938), 921–940.Google Scholar
  94. 94.
    van der Pol, B., & H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral (Cambridge University Press, 1950).Google Scholar
  95. 95.
    van der Pol, B., & K. F. Niessen, On simultaneous operational calculus. Phil. Mag. 7 ser. 11 (1931), 368–376.Google Scholar
  96. 96.
    van der Pol, B., & K. F. Niessen, Symbolic calculus. Phil. Mag. 7 ser. 13 (1932), 537–577.Google Scholar
  97. 97.
    Riemann, B., Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsb. K. Preuss. Acad. Wiss. (Berlin) (1859, 1860), 671–680. Also in Bernard Riemann's gesammelte mathematische Werke (Ed. H. Weber) (Leipzig: Teubner, 1876). (2nd Edition (1892); Reprinted New York: Dover, 1953), 145–153 with editorial notes, 154–155. English translation: On the number of primes less than a given magnitude, by H. M. Edwards as an appendix of his Riemann's Zeta Function (New York and London: Academic Press, 1974), pp. 299–305.Google Scholar
  98. 98.
    Schlesinger, L., Handbuch der Theorie der linearen Differentialgleichungen Bd. 1 (Leipzig: Teubner, 1895; Reprinted New York: Johnson).Google Scholar
  99. 99.
    Schlesinger, L., Sur l'intégration des équations slinéaire à l'aide des intégrales définies. C. R. Acad. Sci. (Paris) 120 (1895), 1396–1398.Google Scholar
  100. 100.
    Schlesinger, L., Ueber die Integration linearer homogener Differentialgleichungen durch Quadraturen. J. reine ang. Math. (Crelle) 116 (1896), 97–132.Google Scholar
  101. 101.
    Schlesinger, L., Handbuch der Theorie der linearen Differentialgleichungen Bd. 2 (Leipzig: Teubner, 1897 and 1898; Reprinted New York: Johnson.)Google Scholar
  102. 102.
    Schlesinger, L., Bericht über die Entwicklung der Theorie der linearen Differentialgleichungen seit 1865. Jahresb. Deutsche. Math. Ver. 18 (1909), 133–266.Google Scholar
  103. 103.
    Schlesinger, L., Einfuhrung in die Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage (Berlin: de Gruyter, 1922).Google Scholar
  104. 104.
    Schlissel, A., The development of asymptotic solutions of linear ordinary differential equations, 1817–1920. Arch. Hist. Ex. Sci. 16 (1977), 307–378.Google Scholar
  105. 105.
    Schlömilch, O., Review of Studien uber die Integration linearer Differentialgleichungen by S. Spitzer, Z. Math. Phys. (Schlömilch) 5 (1860), Literaturzeitung 17–18.Google Scholar
  106. 106.
    Spitzer, S., Transformation der Function \(x^n e^{\lambda x^2 } \). Arch. Math. Phys. (Grunert) 58 (1876), 431–432.Google Scholar
  107. 107.
    Tamarkin, J. D., On Laplace's integral equations. Trans. Amer. Math. Soc. 28 (1926), 417–425.Google Scholar
  108. 108.
    Tchebycheff, P., Sur deux théorèmes relatifs aux probabilités. (Translation from a Russian original, dated 1887, by I. Lyon) Acta Math. 14 (1891), 305–315.Google Scholar
  109. 109.
    Titchmarsh, E. C., Hankel transforms. Proc. Camb. Phil. Soc. 21 (1923), 463–473.Google Scholar
  110. 110.
    Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals (Oxford: Clarendon, 1937).Google Scholar
  111. 111.
    Tricomi, F. G., Article on Pincherle in Dictionary of Scientific Biography Vol. X. (Ed.. C. C. Gillespie et al.). (New York: Scribner, 1974), 610.Google Scholar
  112. 112.
    Volterra, V., Leçons sur les fonctions de lignes (Ed. J. Pérès) (Paris: Gauthier-Villars, 1913) (Coll. de monographies sur la théorie des fonctions). See, in particular, Chapter 9.Google Scholar
  113. 113.
    Wagner, K. W., Über eine Formel von Heaviside zur Berechnung von Einschaltvorgängen. (Mit Anwendungsbeispielen.) Arch. Electrotech. 4 (1916), 159–193.Google Scholar
  114. 114.
    Widder, D. V., The Laplace Transform (Princeton University Press, 1946).Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Michael A. B. Deakin
    • 1
  1. 1.Mathematics DepartmentMonash UniversityClaytonAustralia

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