Advertisement

Archive for History of Exact Sciences

, Volume 68, Issue 5, pp 547–597 | Cite as

Integral equations between theory and practice: the cases of Italy and France to 1920

  • T. Archibald
  • R. Tazzioli
Article

Abstract

In 1899, Ivar Fredholm discovered how to treat an integral equation using conceptual methods from linear algebra and use these ideas to solve certain classes of boundary value problems. He formulated a theory allowing him both to unify large classes of problems and to attack several problems fruitfully. The historical literature on the theory of integral equations has concentrated largely on the unification that was afforded by Hilbert and his school, but has not throughly investigated the roots of the subject in the older theory of partial differential equations, as developed for instance by Fredholm himself but also by Volterra and Levi-Civita. By concentrating on work issuing from this older tradition, in particular on French and Italian work, the paper shows how the new theory of integral equations was enthusiastically received, especially for its fruitful applications to areas of mathematical physics such as hydrodynamics, elasticity, and heat theory.

Keywords

Integral Equation Dirichlet Problem Laplace Equation Betti Biharmonic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Almansi, Emilio. 1896. Sulla integrazione dell’equazione differenziale \(\Delta ^{2} \Delta ^{2}=0\). Atti R. Acc. Scienze di Torino 31: 881–888.zbMATHGoogle Scholar
  2. Almansi, Emilio. 1897. Sulla deformazione della sfera elastica. Memorie R. Acc. Scienze di Torino s. II 47: 103–125.Google Scholar
  3. Almansi, Emilio. 1899a. Sull’integrazione dell’equazione differenziale \(\Delta ^{2n}=0\). Ann. matem. pura ed appl. s. III 2: 1–51.CrossRefzbMATHGoogle Scholar
  4. Atti del IV Congresso Internazionale dei Matematici. Roma, Accademia dei Lincei. 3 v. 1909.Google Scholar
  5. Bernstein, Sergej. 1904. Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre. Math. Annalen 59: 20–76.CrossRefzbMATHGoogle Scholar
  6. Betti, Enrico. 1872–73. Teoria della elasticità. Pisa: Tip. Pieraccini, 1874; Nuovo Cimento s. II, vol. 7–10. Reprinted in (Capecchi, Ruta and Tazzioli 2006).Google Scholar
  7. Bôcher, Maxim. 1909. An introduction to the theory of integral equations. Cambridge: Cambridge University Press.Google Scholar
  8. Boggio, Tommaso. 1900a. Sull’equilibrio delle membrane elastiche piane. Nota I. Atti Acc. Scienze di Torino 35: 219–239; Nota II, Ibid., 239 (1901a); see also Nuovo Cimento s. IV, vol. 12, 1900, 170–190; s. V, vol. 1, 1901, 161–178.Google Scholar
  9. Boggio, Tommaso. 1900b. Sull’equilibrio delle membrane elastiche piane. Atti Acc. Scienze Torino 35: 219–239.Google Scholar
  10. Boggio, Tommaso. 1900c. Un teorema di reciprocità sulle funzioni di Green d’ordine qualunque. Atti Acc. Scienze di Torino 35: 498–509.zbMATHGoogle Scholar
  11. Boggio, Tommaso. 1900d. Integrazione dell’equazione in una corona circolare e in uno strato sferico. Atti Ist. Veneto di Scienze, Lettere ed Arti 59(II): 497–508.Google Scholar
  12. Boggio, Tommaso. 1901a. Integrazione dell’equazione in un’area ellittica. Atti Ist. Veneto di Scienze, Lettere ed Arti 60(II): 591–609.Google Scholar
  13. Boggio, Tommaso. 1901b. Sull’equilibrio delle piastre elastiche incastrate. Rend. R. Acc. dei Lincei s. V 10(I): 197–205.Google Scholar
  14. Boggio, Tommaso. 1904. Sulla deformazione delle piastre elastiche cilindriche di grossezza qualunque. Rend. R. Acc. dei Lincei s. V 13(II): 419–427.Google Scholar
  15. Boggio, Tommaso. 1906. Risoluzione del problema dei valori al contorno per alcuni casi di equazioni alle derivate parziale. Rend. Circolo Mat. Palermo 21: 283–306.CrossRefzbMATHGoogle Scholar
  16. Boggio, Tommaso. 1907a. Nuova risoluzione di un problema fondamentale della teoria dell’elasticità. Rend. R. Acc. dei Lincei 16(II): 248–255.Google Scholar
  17. Boggio, Tommaso. 1907b. Sull’equazione del moto vibratorio delle membrane elastiche. Rend. R. Acc. dei Lincei (5) 16(II): 386–393.Google Scholar
  18. Boggio, Tommaso. 1907c. Integrazione dell’equazione funzionali che reggie la caduta di una sfera in un liquido viscoso. Rend. R. Acc. dei Lincei (5) 16(II): 613–620, 730–737.Google Scholar
  19. Boltzmann, Ludwig. 1874. Theorie der elastischen Nachwirkung. Ber. K. K. Akad. Wien 70: 275–306.Google Scholar
  20. Boltzmann, Ludwig. 1876. Zur Theorie der elastischen Nachwirkung. Pogg Ann. Erg. 7: 624.Google Scholar
  21. Bottazzini, Umberto. 1986. The higher calculus. A history of real and complex analysis from Euler to Weierstrass. New York: Springer.CrossRefzbMATHGoogle Scholar
  22. Brezis, Chaim and F. Browder. 1998. Partial differential equations in the 20th century. Advances in Mathematics 135: 76–144.Google Scholar
  23. Capecchi, Danilo, Ruta Giuseppe, and Rossana Tazzioli. 2006. Enrico Betti:Teoria della Elasticità. Benevento: Hevelius.Google Scholar
  24. Charpentier’, E., É. Ghys, and A. Lesne. 2010. L’heritage scientifique de Poincaré. Paris: Belin . English translation The Scientific Legacy of Poincaré. Providence: AMS.Google Scholar
  25. Cerruti, Valentino. 1882. Ricerche intorno all’equilibrio dei corpi elastici isotropi. Memorie R. Acc. dei Lincei s. III 13: 81–123.Google Scholar
  26. d’Adhémar, Robert. 1908. Sur les équations intégrales de M. Volterra. In (Atti Roma 1909), II, 115–121.Google Scholar
  27. Dieudonné, Jean. 1981. History of functional analysis. Amsterdam: North Holland.zbMATHGoogle Scholar
  28. Du Bois-Reymond, Paul. 1888. Bemerkungen über \(\Delta z = 0\). Jour. f ür Math. 103: 204–229.zbMATHGoogle Scholar
  29. Fischer, Ernst. 1907. Sur la convergence en moyenne. Comptes rendus de l’Académie des Sciences 144: 1022–1024.zbMATHGoogle Scholar
  30. Forsyth, A.R. 1908. On the present condition of partial differential equations of the second order as regards formal integration. In (Atti Roma 1908) I, 87–103.Google Scholar
  31. Fredholm, Ivar. 1899. Sur une classe d’équations aux dérivées partielles. Comptes rendus de l’Académie des Sciences 129: 32–34.zbMATHGoogle Scholar
  32. Fredholm, Ivar. 1900. Sur une nouvelle méthode pour la résolution du problème de Dirichlet. Stockholm Öfver 57: 39–46.Google Scholar
  33. Fredholm, Ivar. 1902. Sur une classe de transformations rationnelles. Comptes rendus de l’Académie des Sciences 134: 219–222.zbMATHGoogle Scholar
  34. Fredholm, Ivar. 1903. Sur une classe d’équations fonctionelles. Acta Math. 27: 365–390.CrossRefzbMATHMathSciNetGoogle Scholar
  35. Fredholm, Ivar. 1905. Solution d’un prolblème fondamentale de la théorie de l’élasticité. Arkiv för Mat. fys, och ast. 2(28): 8Google Scholar
  36. Fubini, Guido. 1905. Un’osservazione sulla teoria delle funzioni poliarmoniche. Rendiconti Istit. Lombardo 38: 449–453.zbMATHGoogle Scholar
  37. Goursat, Édouard. 1907. Sur un cas élémentaire de l’équation de Fredholm. Bulletin de la Soc. math. de France 35: 163–173.Google Scholar
  38. Goursat, Édouard. 1915. Traité d’analyse. 2nd edn., Paris. On Integral equations see v. 3, chaps 30–33, 324–544.Google Scholar
  39. Gray, Jeremy. 2013. Henri Poincaré: A scientific biography. Princeton: Princeton University PressGoogle Scholar
  40. Groetsch, Charles W. 2003. The delayed emergence of regularization theory. Bollettino Storiadelle Scienze Matemetiche 23: 105–120.zbMATHMathSciNetGoogle Scholar
  41. Hadamard, Jacques. 1893. Sur le module maximum qui puisse atteindre un déterminant. Comptes rendus de l’Académie des Sciences 116: 1500–1501; in Oeuvres (vol. 4, Paris, CNRS, 1968) vol. 1, 237–238.Google Scholar
  42. Hadamard, Jacques. 1908. Sur certains cas intéressants du problème biharmonique. Atti del IV Congresso dei Matematici 2: 61–63. Roma: Accademia dei Lincei; in Oeuvres vol. 3, 643–645.Google Scholar
  43. Hadamard, Jacques. 1909. Sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées. Mémoires savants étrang. s. II 33(4); in Oeuvres vol. 2, 515–629. Prix Vaillant.Google Scholar
  44. Hadamard, Jacques. 1910. Sur les ondes liquides Ière et IIéme notes. Comptes rendus de l’Académie des Sciences 150: 609–611, 772–774; in Oeuvres vol. 3, 1301–1303, 1317–1320.Google Scholar
  45. Hadamard, Jacques. 1928. Le développement et le rôle scientifique du calcul fonctionnel. Atti del Congresso Internazionale dei Matematici, Bologna, Zanichelli, Bologna 1929, vol. 1, 143–161; In Oeuvres vol. 1, 435–453.Google Scholar
  46. Happel, J., and H. Brenner. 1965. Low Reynolds number hydrodynamics. Englewood Cliffs NY: Prentica Hall.Google Scholar
  47. Hellinger, Ernst. 1935. Hilberts Arbeiten über Integralgleichungen und unendliche Gleichungssysteme. In (Hilbert Gesammelte Abhandlungen) vol. III, 95–145.Google Scholar
  48. Hellinger, Ernst and Otto Toeplitz. 1927. Integralgleichungen und Gleichungen mit unendlichviele unbekannten. Enzyklopädie der mathematischen Wissenschaften II.3.2, section IIC. 13: 1335–1597.Google Scholar
  49. Heywood, Horace B. 1908. L’équation de Fredholm et quelques-unes de ses applications. Thèse, Université de Paris.Google Scholar
  50. Heywood, Horace B., and Maurice Fréchet. 1912. L’équation de Fredholm et ses application á la physique mathématique. Préface de J. Hadamard. Paris, HermannGoogle Scholar
  51. Hilbert, David. 1900. Sur le principe de Dirichlet. Trans. M. Laugel. Nouvelles Annales de Mathématiques s. III 19: 337–344.Google Scholar
  52. Hilbert, David. 1904. Grundzüge einer allgemeinen Theorie der linearen Integralgkleichung. Gött. Nachr. 49–91: 213–259.Google Scholar
  53. Hilbert, David. 1935. Gesammelte Abhandlungen. Berlin: Springer. Cited here from the Second edition, 1970.Google Scholar
  54. Korn, Arthur. 1899. Lehrbuch der Potentialtheorie, allgemeine Theorie des Potentials und der Potentialfunktionen im Raume. Berlin: F. Dümmler.Google Scholar
  55. Kneser, Arthur. 1907. Die Theorie der Integralgleichungen und die Darstellung willkürlicher Funktionen in der mathematischen Physik. Math. Annalen 63: 477–524.CrossRefzbMATHMathSciNetGoogle Scholar
  56. Korn, Arthur. 1902. Fünf Abhandlungen zur Potentialtheorie. Berlin: F. Dümmler.zbMATHGoogle Scholar
  57. Korn, Arthur. 1907. Sur les équations de l’élasticité. Ann. Sci. Ecole Normale Sup. s. III 24: 9–75.zbMATHMathSciNetGoogle Scholar
  58. Lalesco, Trajan. 1912. Introduction à la théorie des équations intégrales. Préface de E. Picard. Paris, Hermann.Google Scholar
  59. Langlois, W.E. 1964. Slow viscous flow. New York: Macmillan.Google Scholar
  60. Lauricella, Giuseppe. 1895. Equilibrio dei corpi elastici isotropi. Annali R. Scuola Norm. Sup. di Pisa 7: 1–119.MathSciNetGoogle Scholar
  61. Lauricella, Giuseppe. 1895–96. Integrazione dell’equazione in un campo di forma circolare. Atti Acc. Scienze Torino 31: 1010–1018.Google Scholar
  62. Lauricella, Giuseppe. 1896. Sull’equazione delle vibrazioni delle placche elastiche incastrate. Memorie Acc. Scienze Torino s. II 46: 65–92.Google Scholar
  63. Lauricella, Giuseppe. 1898. Integrazione della doppia equazione di Laplace in un campo a forma di corona circolare. Atti Ist. Veneto di Scienze, Lettere ed Arti 57: 236–250.Google Scholar
  64. Lauricella, Giuseppe. 1901a. Sulle funzioni biarmoniche. Rend. R. Acc. dei Lincei s. V 10: 147–150.zbMATHGoogle Scholar
  65. Lauricella, Giuseppe. 1901b. Sulla deformazione di una sfera elastica isotropa per dati spostamenti in superficie. Ann. matem. pura ed appl. s. III 6: 289–299.CrossRefzbMATHGoogle Scholar
  66. Lauricella, Giuseppe. 1903. Sulla deformazione di una sfera elastica isotropa per date tensioni in superficie. Nuovo Cimento s. V 5: 5–26.CrossRefzbMATHGoogle Scholar
  67. Lauricella, Giuseppe. 1904. Sulle formole che danno la deformazione di una sfera elastica isotropo. Rend. R. Acc. dei Lincei s. V 13(II): 583–590.Google Scholar
  68. Lauricella, Giuseppe. 1905. Sull’integrazione delle equazioni dell’equilibrio dei corpi elastici isotropi. Ann. matem. pura ed appl. s. III 11: 269–283.CrossRefGoogle Scholar
  69. Lauricella, Giuseppe. 1907. Alcune applicazioni della teoria delle equazioni funzionali alla fisica matematica. Nuovo Cimento s. V 13: 104–118, 155–174, 237–262, 501–518.Google Scholar
  70. Lauricella, Giuseppe. 1908a. Sulla vibrazione delle placche elastiche incastrate. Rend. R. Acc. dei Lincei s. V 17(II): 193–204.Google Scholar
  71. Lauricella, Giuseppe. 1908b. Sopra alcune equazioni integrali. Rend. R. Acc. dei Lincei s. V 17(I): 775–786.Google Scholar
  72. Lauricella, Giuseppe. 1908c. Applicazione della teoria di Fredholm al problema del raffreddamento dei corpi. Ann. matem. pura ed appl. s. III 14: 143–169.CrossRefGoogle Scholar
  73. Lauricella, Giuseppe. 1908d. Sull’equazione \(\triangle ^{2i}V=0\) e su alcune estensioni delle equazioni dell’equilibrio dei corpi elastici isotropi. In (Atti Roma 1908d) vol. III, 33–59.Google Scholar
  74. Lauricella, Giuseppe. 1909a. Sur l’intégration de l’équation relative à l’équilibre des plaques élastiques encastrées. Acta Mathematica 32: 201–256.CrossRefzbMATHMathSciNetGoogle Scholar
  75. Lauricella, Giuseppe. 1909b. Sull’equazione integrale di prima specie. Rend. R. Acc. dei Lincei s. V 18(II): 71–75.Google Scholar
  76. Lauricella, Giuseppe. 1911. Sulla risoluzione dell’equazione integrale di prima specie. Rend. R. Acc. dei Lincei s. V 20: 528–536.zbMATHGoogle Scholar
  77. Lauricella, Giuseppe. 1912. Sulla chiusura di sistemi di funzioni ortogonali e dei nuclei delle funzioni integrali. Rend. R. Acc. dei Lincei s. V 21(I): 675–685.Google Scholar
  78. Lebesgue, Henri. 1904. Leçons sur l’intégration et la recherche de fonctions primitives. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  79. Le Roux, Jean Marie. 1895. Sur les intégrales des équations linéaires. Ann. Sci. ENS s. III 12: 227–316.Google Scholar
  80. Le Roux, Jean Marie. 1903. Recherches sur les équations aux dérivées partielles. Journal de mathématiques pures et appliquées S. V. 9: 403–455.Google Scholar
  81. Levi, Eugenio Elia. 1906. Su un lemma del Poincaré. Rend. R. Acc. dei Lincei s. V 14: 83–89, 353–358.Google Scholar
  82. Levi, Eugenio Elia. 1907. Sulle equazioni integrali. Rend. R. Acc. dei Lincei s. V 16(II): 604–6012.Google Scholar
  83. Levi, Eugenio Elia. 1908. Sur l’application des équations intégrales au problème de Riemann. Gött. Nachr., 249–252.Google Scholar
  84. Levi-Civita, Tullio. 1954–1973. Opere Matematiche vol. 6. Zanichelli: Bologna.Google Scholar
  85. Levi-Civita, Tullio. 1895–96. Sull’inversione degli integrali definiti nel campo reale. Atti Acc. Scienze di Torino 31: 25–51; In (Levi-Civita 1954–1973) vol. 1, 159–184.Google Scholar
  86. Levi-Civita, Tullio. 1897–98. Sopra una trasformazione in se stessa dell’equazione \(\Delta _2 \Delta _2=0\). Atti Ist. Veneto di Scienze, Lettere ed Arti 56: 1399–1410; In (Levi-Civita 1954–1973) vol. 1, 357–367.Google Scholar
  87. Levi-Civita, Tullio. 1898. Sulla integrazione dell’equazione \(\Delta _2 \Delta _2=0\). Atti Acc. Scienze di Torino 33: 932–956; In (Levi-Civita 1954–1973) vol. 1, 331–355.Google Scholar
  88. Liapunov, Alexander Mikhaylovich. 1898. Sur certaines questions qui se rattachent au problème de Dirichlet. Journ. de math. pures et appl. s. V 4: 241–311.Google Scholar
  89. Lützen, Jesper. 1990. Joseph Liouville 1809–1882: Master of pure and applied mathematics. New York: Springer.CrossRefzbMATHGoogle Scholar
  90. Marcolongo, Roberto. 1903. Teoria matematica della elasticità: lezioni dettate nella R. Università di Messina, anno scolastico 1902–1903. Messina.Google Scholar
  91. Marcolongo, Roberto. 1907. La teoria delle equazioni integrali e le sue applicazioni alla Fisica matematica. Rend. R. Acc. dei Lincei s. V 6(I): 742–749.Google Scholar
  92. Mason, Charles Max. 1903. Randwertaufgaben bei gewöhnlichen Differentialgleichungen. Ph. D. thesis, Göttingen.Google Scholar
  93. Mathieu, Emile. 1869. Mémoire sur les équations aux différences partielles du quatrième ordre \(\Delta \Delta u = 0\) et sur l’équilibre d’élasticité d’un corps solide. Journ. Math. pures et appliquées, s. II 14: 378–421.Google Scholar
  94. Mawhin, Jean. 2006/2010. Henri Poincaré and the partial differential equations of mathematical physics. In Charpentier, Ghys, and Lesne, 257–278.Google Scholar
  95. Mazliak, Laurent, and Rossana Tazzioli. 2009. Mathematicians at war: Volterra and his French colleagues in World War I. New York: Springer.CrossRefGoogle Scholar
  96. Maz’ya, Vladimir, and Tatyana Shaposhnikova. 1998. Jacques Hadamard, a universal mathematician. Providence: American Mathematical Society.zbMATHGoogle Scholar
  97. Meleshko, V.V. 2003. Selected Topics in the history of the two-dimensional biharmonic problem. Applied Mechanics Reviews 56: 33–85.CrossRefGoogle Scholar
  98. Monna, A.F. 1975. Dirichlet’s principle: A mathematical comedy of errors. Oosthoek, Scheltena & Hoeltema.Google Scholar
  99. Nabonnand, Philippe (ed.). 1999. La correspondance entre Henri Poincaré et Gösta Mittag-Leffler. Basel: Birkhäuser.zbMATHGoogle Scholar
  100. Nastasi, Pietro, and Rossana Tazzioli. 2004. Sulla determinazione della m.ma funzione di Green e questioni connesse. Rend. Circ. Matem. Palermo Suppl. s. II 74: 71–101.MathSciNetGoogle Scholar
  101. Nastasi, Pietro, and Rossana Tazzioli. 2006. Problems of method in Levi-Civita’s contributions to hydrodynamics. Revue d’histoire des mathématiques 12: 81–118.zbMATHMathSciNetGoogle Scholar
  102. Orlando, Luciano. 1906. Sull’integrazione della \(\Delta _4\) in un parallelepipedo rettangolo. Rend. Circolo Matem. Palermo 21: 316–344.CrossRefzbMATHGoogle Scholar
  103. Orlando, Luciano. 1908. Sulla risoluzione delle equazioni integrali. In (Atti Roma 1908) II, 1908, 122–128.Google Scholar
  104. Osgood, William Fogg. 1919. The life and services of Maxime Bôcher. Bulletin of the American Mathematical Society 25: 337–350.CrossRefzbMATHMathSciNetGoogle Scholar
  105. Painlevé, Paul. 1909. Mécanique: Le genèse de la mécanique et ses influences sur les autres sciences. De la méthode dans les sciences, par MM. les professeurs Bouasse [et al]. Paris, F. Alcan, 363–409.Google Scholar
  106. Pedersen, Gert K. 1988. Analysis Now. New York: Springer.Google Scholar
  107. Picard, Émile. 1893. Traité d’Analyse, vol. 3. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  108. Picard, Émile. 1904a. Sur certaines solutions doublement périodiques de quelques équations aux dérivées partielles. Comptes rendus de l’Académie des Sciences vol. 138, 181–183; in (Picard 1981) vol. 2, 687–689.Google Scholar
  109. Picard, Émile. 1904b. Sur une équation fonctionelle. Comptes rendus de l’Académie des Sciences 139: 245–248; in (Picard 1981) vol. 4, 313–316.Google Scholar
  110. Picard, Émile. 1906a. Sur quelques problèmes de Physique mathématique se rattachant à l’équation de M. Fredholm. Comptes rendus de l’Académie des Sciences 142: 861–865; in (Picard 1981) vol. 4, 317–321.Google Scholar
  111. Picard, Émile. 1906b. Sur quelques applications de l’équation fonctionelle de M. Fredholm. Rendiconti Circolo matem. Palermo 22: 1906, 241–259; in (Picard 1981), vol. 4, 323–341.Google Scholar
  112. Picard, Émile. 1907. La mécanique classique et ses approximations successives. Rivista di Scienza 1(1): 14.Google Scholar
  113. Picard, Émile. 1908. La mathématique dans ses rapports avec la physique, in (Atti Roma 1908), I, 183–195.Google Scholar
  114. Picard, Émile. 1909a. Quelques remarques sur les équations intégrales de première espèce et sur certains problèmes de Physique mathématique. Comptes rendus de l’Académie des Sciences 148: 1909, 1563–1568; in (Picard 1981) vol. 4, 379–384.Google Scholar
  115. Picard, Émile. 1909b. Sur les équations intégrales de première espèce. Comptes rendus de l’Académie des Sciences 148: 1707–1708; in (Picard 1981) vol. 4, 385–386.Google Scholar
  116. Picard, Émile. 1910. Sur un théoréme général relatif aux équations intégrales de première espèce et sur quelques problémes de physique mathématique. Rendiconti Circolo Matem. Palermo 29: 79–97.CrossRefzbMATHGoogle Scholar
  117. Picard, Émile. 1981. Œuvres de Charles-Émile Picard, vol. 4. Paris: CNRS.Google Scholar
  118. Picciati, Giuseppe. 1907. Integrazione dell’equazione funzionali che reggie la caduta di una sfera in un liquido viscoso. Rend. R. Acc. dei Lincei (5) 16(II): 45–50.Google Scholar
  119. Plancherel, M. 1912. La théorie des équations intégrales. Enseignement mathématique 14: 89–107.zbMATHGoogle Scholar
  120. Poincaré, Henri. 1913–1965. Œuvres, vol. 11. Paris: Gauthier-Villars.Google Scholar
  121. Poincaré, Henri. 1890. Sur les équations aux dérivées partielles de la physique mathématique. American Journal of Mathematics 1: 211–294; in (Poincaré 1913–1965) vol. 9, 28–113.Google Scholar
  122. Poincaré, Henri. 1894. Sur les équations de la physique mathématique. Rendiconti Circ. Matem. Palermo 8: 57–156; in (Poincaré 1913–1965) vol. 9, 123–196.Google Scholar
  123. Poincaré, Henri. 1897. La méthode de Neumann et le problème de Dirichlet. Acta Math 20: 59–142; in (Poincaré 1913–1965) vol. 9, 202–272.Google Scholar
  124. Poincaré, Henri. 1908a. Remarques sur l’équation de Fredholm. Comptes rendus de l’Académie des Sciences 147: 1367–1371; In (Poincaré 1913–1965) vol. 3, 540–544.Google Scholar
  125. Poincaré, Henri. 1908b. Sur quelques applications de la méthode de Fredholm. Comptes rendus de l’Académie des Sciences 148: 125–126; In (Poincaré 1913–1965) vol. 3, 545–546.Google Scholar
  126. Poincaré, Henri. 1908c. L’avenir des mathématiques, in (Atti Roma 1908), vol. I, 167–182.Google Scholar
  127. Riesz, Friedrich. 1907. Ueber orthogonale Funktionsysteme, Gött. Nachr., 116–122.Google Scholar
  128. Rouché, Eugène. 1860. Sur le calcul inverse des intégrales définis. Comptes rendus de l’Académie des Sciences 51: 126–128.Google Scholar
  129. Schmidt, Erhard. 1905. Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener Funktionen. Göttingen.Google Scholar
  130. Schmidt, Erhard. 1907. Zur Theorie der linearen und nichtlinearen Integralgleichungen. Math. Annalen 63: 433–476.CrossRefzbMATHGoogle Scholar
  131. Schwarz, Hermann Amandus. 1885. Ueber ein die Flächen kleinsten Flächeninhalts betreffendes Problem der Variationsrechnung. Acta Soc. Sci. Finn. 15: 315–362. In Festschrift zum siebzigsten Geburtstage der Herrn Karl Weierstrass. Helsingfors, Druck der Finnischen Literatur-GesellschaftGoogle Scholar
  132. Sherman, D.I. 1940. On the solution of a plane static problem of the theory of elasticity under prescribed forces. Dokl. Akad. Nauk SSSR 28: 25–28.zbMATHGoogle Scholar
  133. Siegmund-Schultze, Reinhard. 1982. Die Anfänge der Funktionalanalysis und ihr Platz im Umwälzungsprozeß der Mathematik um 1900. Arch. Hist. Exact Sciences 26: 13–71.zbMATHMathSciNetGoogle Scholar
  134. Siegmund-Schultze, Reinhard. 2003. The origins of functional analysis. In A history of analysis, ed. H.N. Jahnke, 385–407. Providence: American Mathematical Society.Google Scholar
  135. Smithies, Frank. 1965. Integral equations. Cambridge: Cambridge University Press.Google Scholar
  136. Somigliana, Carlo. 1890. Sulle equazioni della elasticità. Ann. matem. pura ed appl. s. II 17: 37–64.Google Scholar
  137. Somigliana, Carlo. 1894. Sopra gli integrali delle equazioni della isotropia elastica. Nuovo Cimento s. III 36(29–39): 113–126.CrossRefGoogle Scholar
  138. Stekloff, Vladimir. 1900. Mémoire sur les fonctions harmoniques de M. Poincaré. Toulouse Ann. s. II 2: 273–303.CrossRefMathSciNetGoogle Scholar
  139. Stekloff, Vladimir. 1902. Sur les problèmes fondamentaux de la physique mathématique. Annales de l’Ec. Norm. Sup. s. III 19: 191–259, 455–490.Google Scholar
  140. Stokes, George Gabriel. 1850. On the effect of the internal friction of fluids on the motion of pendulums. Transactions Cambridge Philosophical Society 9: 8–106.Google Scholar
  141. Tazzioli, Rossana. 1994. Il teorema di rappresentazione di Riemann: Critica e Interpretazione di Schwarz. Rend. Circolo Matem. Palermo Suppl. s. II 34: 95–132.MathSciNetGoogle Scholar
  142. Tazzioli, Rossana. 2001. Green’s Function in some contributions of 19th century mathematicians. Historia Mathematica 28: 232–252.CrossRefzbMATHMathSciNetGoogle Scholar
  143. Tedone, Orazio. 1907. Sui metodi della fisica matematica. Atti del Primo Congresso della S.I.P.S., 33–47. Parma.Google Scholar
  144. Tricomi, Francesco. 1957. Integral equations. New York: Interscience. Page references here from the 1985 reprint by Dover, New York.Google Scholar
  145. Volterra, Vito. 1954–1960. Opere matematiche, vol. 4. Roma: Bardi.Google Scholar
  146. Volterra, Vito. 1884. Sopra un problema di elettrostatica. Nuovo Cimento s. III 16: 49–57; in (Volterra 1954–1960) vol. 1, 188–195.Google Scholar
  147. Volterra, Vito. 1895. Sopra un sistema di equazioni differenziali. Atti R. Acc. Scienze di Torino 30: 445–454; in (Volterra 1954–1960) vol. 2, 122–128.Google Scholar
  148. Volterra, Vito. 1896a. Sulla inversione degli integrali definiti. Atti R. Acc. Scienze di Torino 31: 311–323, 400–408, 557–567, 693–708; in (Volterra 1954–1960) vol. 2, 216–254.Google Scholar
  149. Volterra, Vito. 1896b. Sulla inversione degli integrali definiti. Rend. R. Acc. dei Lincei s. V 5: 177–185; in (Volterra 1954–1960) vol. 2, 255–262.Google Scholar
  150. Volterra, Vito. 1896c. Sulla inversione degli integrali multipli. Rend. R. Acc. dei Lincei s. V 6: 289–300; in (Volterra 1954–1960) vol. 2, 263–275.Google Scholar
  151. Volterra, Vito. 1898. Sul fenomeno delle seiches. Conf. tenuta al Congr. della Soc. italiana di Fisica in Torino il 23 settembre 1892. Nuovo Cimento s. IV 8: 270–272; in (Volterra 1954–1960) vol. 2, 370–372.Google Scholar
  152. Volterra, Vito. 1901–02. Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali. Ann. R. Univ. di Roma, 3–28; in (Volterra 1954–1960) vol. 3, 14–29.Google Scholar
  153. Volterra, Vito. 1906. Leçons sur l’intégration des équations différentielles aux dérivées partielles; in (Volterra 1954–1960) vol. 3, 63–141; publ. also Paris, Hermann, 1912.Google Scholar
  154. Volterra, Vito. 1912. L’applicazione del calcolo ai fenomeni di eredità, French version Revue du Mois, 556–575 and Italian version in Saggi Scientifici, 1920, Bologna, Zanichelli, 189–218; in (Volterra 1954–1960) vol. 3, 554–568.Google Scholar
  155. Volterra, Vito. 1913a. Leçons sur les équations intégrales et les équations intégro-différentielles. Paris: Gauthier-Villars.zbMATHGoogle Scholar
  156. Volterra, Vito. 1913b. Leçons sur les fonctions de lignes. Paris: Gauthier-Villars.Google Scholar
  157. Volterra, Vito, and Joseph Përès. 1936. Théorie générale des fonctionelles. Paris: Gauthier-Villart.Google Scholar
  158. Weyl, Hermann. 1908. Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems. Göttingen.Google Scholar
  159. Weyl, Hermann. 1909. Ueber die Konvergenz von Reihen, die nach Orthogonalfunktionen fortschreiten. Math. Annalen 67: 225–245.CrossRefzbMATHMathSciNetGoogle Scholar
  160. Zaremba, Stanislaw. 1901. Ueber die Laplacesche Gleichung und die Methoden von Neumann und Robin. Krak. Anz, 171–189.Google Scholar
  161. Zaremba, Stanislaw. 1902. Sur l’intégration de l’équation \(\Delta u+u=0\). Journ. de Math. pures et appl. s. V 8: 59–117.zbMATHGoogle Scholar
  162. Zaremba, Stanislaw. 1905. Solution générale du problème de Fourier. Krak. Anz, 69–168.Google Scholar
  163. Zeilon, Nils. 1930. Biographie de Ivar Fredholm. Acta Mathematica 54: I–XVI.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.UFR de mathématiques, Laboratoire Paul Painlevé, UMR CNRS 8524Université de Sciences et Technologie de LilleLilleFrance

Personalised recommendations