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A Guide to Tables of Bessel Functions


Authors: Harry Bateman and Raymond Clare Archibald
Journal: Math. Comp. 1 (1944), 205-308
DOI: https://doi.org/10.1090/S0025-5718-1944-0011175-4
Corrigendum: Math. Comp. 10 (1956), 262-263.
Corrigendum: Math. Comp. 3 (1948), 332.
Corrigendum: Math. Comp. 2 (1947), 320.
Corrigendum: Math. Comp. 2 (1947), 228.
Corrigendum: Math. Comp. 2 (1946), 196.
Corrigendum: Math. Comp. 2 (1946), 148.
Corrigendum: Math. Comp. 2 (1946), 95-96.
Corrigendum: Math. Comp. 2 (1946), 63-64.
Corrigendum: Math. Comp. 1 (1945), 460.
Corrigendum: Math. Comp. 1 (1945), 432.
Corrigendum: Math. Comp. 1 (1945), 408.
Corrigendum: Math. Comp. 1 (1945), 375.
MathSciNet review: 0011175
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References | Additional Information

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  • [1] F. W. Bessel came upon these functions in discussion of planetary perturbations, and made the first tables of $ {J_0}(x)$ and $ {J_1}(x)$; Akad. d. Wissen., Berlin, Math. Klasse, Abh. for the year 1824, Berlin, 1826. The name ``Bessel's function'' and the notation $ {J_n}(x)$ are due to O. Schlömilch, Z. Math. Phys., v. 2, 1857.
  • [2] D. Bernoulli, Akad. Nauk. Leningrad, Commentarii, v. 6 for the year 1732 and 1733, 1738, p. 116-119. See also G. A. Maggi, Accad. d. Lincei, Atti, s. 3, Transunti, v. 4, 1880, p. 259-263; and M. Bôcher, New York Math. So., Bull., v. 2, 1893, p. 107-109.
  • [1] H. Weber, J.f. d. reine u. angew. Math., v. 76, 1873, p. 9; Math. Annalen, v. 6, 1873, p. 148.
  • [2] C. G. Neumann, Theorie der Bessel'schen Funktionen, Leipzig, 1867, p. 41.
  • [3] H. Hankel, Math. Annalen, v. 1, 1869, p. 471.
  • [4] L. Euler, ``De perturbatione motus chordarum ab earum pondere oriunda,'' Akad. Nauk, Acta, for 1781, St. Petersburg, 1784, p. 187. Compare M. Bôcher, New York, Math. So., Bull., v. 2, 1893, p. 108-109. In his Institutionum Calculi Integralis, v. 2, St. Petersburg, 1769, p. 191-192, Euler gave also the complete solution of $ {x^{3/2}}y'' + ay = 0$; solutions of this equation are $ {x^{\frac{1}{2}}}{J_2}(4{a^{\frac{1}{2}}}{x^{1/4}}),\pi {x^{\frac{1}{2}}}{Y_2}(4{a^{\frac{1}{2}}}{x^{1/4}})$. See Watson 3, p. 62.
  • [1] A. B. Basset, A Treatise on Hydrodynamics, with numerous Examples, 2v, Cambridge, 1881, v. 2, p. 15.
  • [2] J. Ivory, R. So. London, Trans., v. 113, 1823, p. 409, 495; and v. 128, 1838, part 2, p. 170-229.
  • [3] E. E. Kummer, J.f. d. reine u. angew. Math., v. 12, 1834, p. 144-147; and v. 17, 1837, p. 210-242.
  • [4] N. Nielsen, Handbuch der Theorie der Cylinderfunktionen, Leipzig, 1904, p. 16.
  • [1] P. Schafheitlin, Berlin Math. So., Sitzungsb., v. 8, 1909, p. 64.
  • [1] G. H. Hardy, ``On certain definite integrals considered by Airy and Stokes,'' Quart. J. Math., v. 41, 1910, p. 226-240.
  • [1] W. K. Clifford, Mathematical Papers, London, 1882, p. 346-349.
  • [2] G. Greenhill, Phil. Mag., s. 6, v. 38, 1919, p. 501-528.
  • [1] P. R. Ansell & R. A. Fisher, ``Note on the numerical calculation of a Bessel function derivative,'' London Math. So., Proc., s. 2, v. 24, Records, June 11, 1925, p. iii-v.
  • [1] E. Lommel, Math. Annalen, v. 9, 1876, p. 425-444.
  • [2] J. W. Nicholson, Quar. J. Math., v. 42, 1911, p. 216-224.
  • [3] N. Nielsen, Handbuch der Theorie der Cylinderfunktionen, Leipzig, 1904.
  • [4] L. Schläfli, Math. Annalen, v. 10, 1876, p. 137-142.
  • [1] See R. C. Archibald, ``Euler integrals and Euler's spiral--sometimes called Fresnel integrals and the clothoïde or Cornu's spiral,'' Am. Math. Mo., v. 25, 1918, p. 276-282. (On p. 280, 1. 2 and 14, for Peters, read Gilbert.) Some tables not listed in the present text are there indicated.
  • [1] Kelvin, ``Ether, electricity and ponderable matter,'' So. Electrical Telegraph Engineers, J., v. 18, 1889, p. 4-37; also in Mathematical and Physical Papers, v. 3, London, Cambridge Univ. Press, 1890, p. 484-515.
  • [2] C. S. Whitehead, ``On a generalisation of the functions ber x, bei x, ker x, kei x,'' Quart. J. Math., v. 42, 1911, p. 316-342.
  • [3] J. B. Costello, ``Bessel product functions,'' Phil. Mag., s. 7, v. 21, 1936, p. 308-318.
  • [1] A. Cauchy, Oeuvres, s. 1, v. 1, Paris, 1887, p. 277-278.
  • [1] T. J. I'A. Bromwich, Infinite Series, London, Macmillan, 1908, p. 322-338. For the history of the Euler-Maclaurin sum formula see E. W. Barnes. London Math. So. Proc., s. 2, v. 3, 1905, p. 253-272.
  • [2] H. Poincaré, Acta Math., v. 8, 1886, p. 296
  • [3] C. G. J. Jacobi, Ast. Nach., v. 28, 1848, col. 94; Werke, v. 7, p. 174.
  • [4] S. D. Poisson, École Polytechnique, Paris, J., v. 12, 1823, p. 349.
  • [5] A. Cauchy, Inst, de France, Acad. d. sci., Mémoires divers savans, v. 1, 1827, p. 272; Oeuvres, s. 1, v. 1, p. 277-278.
  • [6] E. W. Barnes, London Math. So., Proc., s. 2, v. 5, 1907, p. 59-116.
  • [7] J. Ivory, R. So. London, Trans., v. 113, 1823, p. 409-495, v. 128, 1838, p. 170-229.
  • [8] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge 1922, p. 207-210.
  • [9] N. S. Koshliakov, London Math. So., J., v. 4, 1929, p. 297-299.
  • [10] B. Strogonoff, Inst. Math. Stekloff, Travaux, v. 9, 1935, p. 223-233.
  • [11] J. R. Airey, Archiv. Math. Phys., v. 22, 1914, p. 30, B.A.A.S., Report, 1913 p. 115. Phil. Mag., s. 7, v. 24, 1937, p. 521-552.
  • [12] A. Lodge, B.A.A.S., Report, 1906, p. 494-498, 1907, p. 95-97.
  • [13] N. Nielsen, Handbuch der Theorie der Cylinderfunktionen, Leipzig, 1904.
  • [14] D. Burnett, Cambridge Phil. So., Proc., v. 26, 1930, p. 145-151.
  • [15] L. V. King, R. So. London, Trans., v. 214A, 1914, p. 373-432.
  • [16] K. Weierstrass, Akad. d. Wissen. Berlin, Sitzungsb. 1885, p. 633-639, 789-805, Werke v. 3, Berlin, 1903, p. 1-37.
  • [17] G. G. Stokes, Cambridge Phil. So. Trans., v. 10, 1857, p. 106-128, Math. & Phys. Papers, v. 4, p. 77-109.
  • [18] Hermann Hankel, Die Cylinderfunctionen erster und zweiter Art, Math. Ann. 1 (1869), no. 3, 467–501 (German). MR 1509635, https://doi.org/10.1007/BF01445870
  • [19] C. Lanczos, J. Math. Phys., M.I.T., v. 17, 1938, p. 123-199.
  • [29] J. Liouville, J. de Math., s. 1, v. 2, 1837, p. 16-35.
  • [21] Adolf Kneser, Einige Sätze über die asymptotische Darstellung von Integralen linearer Differentialgleichungen, Math. Ann. 49 (1897), no. 3-4, 383–399 (German). MR 1510969, https://doi.org/10.1007/BF01444360
  • [22] J. Horn, Ueber eine lineare Differentialgleichung zweiter Ordnung mit einem willkürlichen Parameter, Math. Ann. 52 (1899), no. 2-3, 271–292 (German). MR 1511055, https://doi.org/10.1007/BF01476159
  • [23] George D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc. 9 (1908), no. 2, 219–231. MR 1500810, https://doi.org/10.1090/S0002-9947-1908-1500810-1
  • [24] E. W. Hobson, London Math. So. Proc., s. 2, v. 6, 1908, p. 374-388.
  • [25] G. Hoheisel, J.f.d. reine u. angew. Math., v. 153, 1924, p. 228-244.
  • [26] Y. Ikeda, Tôhoku Math. J., v. 29, 1928, p. 284-290.
  • [27] H. Jeffreys, London Math. So., Proc., s. 2, v. 23, 1925, p. 428-436, Phil. Mag., s. 7, v. 33, 1942, p. 451-456.
  • [28] L. Brillouin, J. de Phys., le Radium, v. 7, 1926, p. 353-368, Inst, de France, Acad. des Sci. Comptes Rendus, v. 183, 1926, p. 24-26.
  • [29] H. A. Kramers, Z.f. Physik, v. 39, 1926, p. 828-840.
  • [30] G. Wentzel, Z.f. Physik, v. 38, 1926, p. 518-529.
  • [31] S. Goldstein, London Math. So., Proc., v. 28, 1928, p. 81-101, v. 33, 1932, p. 246-252.
  • [32] G. N. Watson, R. So. London, Proc., v. 94A, 1918, p. 190-206.
  • [33] F. Carlini, Ricerche sulla convergenza della serie che serva alla soluzione del problema di Keplero, Milan, 1817, 48 p. Translation into German in Jacobi's Werke, v. 7, p. 184-245.
  • [34] C. G. J. Jacobi, Ast. Nach., v. 28, 1848, col. 257-270, Werke, v. 7, p. 175-188.
  • [35] P. Debye, Näherungsformeln für die Zylinderfunktionen für große Werte des Arguments und unbeschränkt veränderliche Werte des Index, Math. Ann. 67 (1909), no. 4, 535–558 (German). MR 1511547, https://doi.org/10.1007/BF01450097
  • [36] G. N. Watson, Cambridge Phil. So., Proc., v. 19, 1916-1919, p. 42-48, 96-110; Phil. Mag., s. 6, v. 35, 1918, p. 369; Bessel Functions, Ch. 8.
  • [37] S. C. van Veen, Asymptotische Entwicklung der Besselschen Funktionen bei großem Parameter und großem Argument, Math. Ann. 97 (1927), no. 1, 696–710 (German). MR 1512384, https://doi.org/10.1007/BF01447890
  • [38] F. Pollaczek, Annalen d. Physik, s. 5, v. 2, 1929, p. 991-1011.
  • [39] R. E. Langer, Am. Math. So., Trans., v. 33, 1931, p. 23-64, v. 34, 1932, p. 447-480.
  • [49] E. Jahnke & F. Emde, Tables of Functions, 3rd. edition, Leipzig & Berlin, 1938.
  • [41] F. Emde, Zeit, angew. Math. Mech., v. 17, 1937, p. 324-340.
  • [42] F. Emde & R. Rühle, Deutsch. Math. Verein, Jahresb., v. 43, 1934, p. 251-270.
  • [43] J. R. Airey, Phil. Mag., s. 7, v. 24, 1937, p. 521-552.
  • [44] J. W. Nicholson, Phil. Mag., s. 6, v. 14, 1907, p. 697-707; v. 16, 1908, p. 271-275; see also A. Erdélyi, Ćasopis Mat. a Fys., v. 67, 1937, p. 240-248.
  • [45] Rayleigh, Phil. Mag., s. 6, v. 20, 1910, p. 1001-1004, Scientific Papers, v. 5, p. 617-620.
  • [46] C. S. Meijer, Akad. v. Wetensch., Amsterdam, Proc., v. 35, 1932, p. 656-667, 853-866, 948-968, 1079-1090, 1291-1303; Math. Ann., v. 108, 1933, p. 321-359.
  • [47] V. A. For, Acad. d. sci., U.R.S.S., Comptes Rendus, new series, v. 1, 1934, p. 99-102.
  • [48] A. Svetlov, Acad. d. sci., U.R.S.S., Comptes Rendus, new series, v. 1, 1934, p. 445-448.
  • [49] S. C. Van Veen, Zentralblatt f. Math., v. 8, 1934, p. 259.
  • [50] J. G. Van der Corput, Composilio Math., v. 1, 1934, p. 15-38, v. 3, 1936, p. 328-372.
  • [51] J. Bijl, Nieuw Archief v. Wiskunde, v. 19, 1936, p. 63-85, Diss. Groningen, 1937, 106p.
  • [52] D. H. Lehmer, MTAC, v. 1, No. 5, 1944, p. 133-135.
  • [53] J. Hadamard, Sur l’expression asymptotique de la fonction de Bessel, Bull. Soc. Math. France 36 (1908), 77–85 (French). MR 1504595
  • [54] W. F. Kibble, A Bessel function in terms of incomplete Gamma functions, J. Indian Math. Soc. 3 (1939), 271–294. MR 0001406
  • [55] G. G. Stokes, Cambridge Phil. So., Trans., v. 9, 1850, p. 184, 186. Math. & Phys. Papers, v. 2, Cambridge, 1883, p. 353, 355.
  • [56] J. McMahon, Annals Math., s. 1, v. 9, 1895, p. 23-30.
  • [57] J. R. Airey, Phys. So. London, Proc., v. 23, 1911, p. 221.
  • [58] W. G. Bickley and J. C. Miller, Note on the reversion of a series, Philos. Mag. (7) 34 (1943), 35–36. MR 0007798
  • [59] H. Bateman, National Math. Mag., v. 18, 1943, p. 10-11.
  • [60] L. Sasaki, Tôhoku Math. J., v. 5, 1914, p. 45-47.
  • [61] J. Fischer, Ingenieur Archiv, v. 10, 1939, p. 95-112.
  • [62] J. R. Airey, B.A.A.S., Report, 1927, p. 253-254.
  • [63] William Marshall, On a new method of computing the roots of Bessel’s functions, Ann. of Math. (2) 11 (1910), no. 4, 153–160. MR 1502404, https://doi.org/10.2307/1967132
  • [64] D. B. Smith, L. M. Rodgers, and E. H. Traub, Zeros of Bessel functions, J. Franklin Inst. 237 (1944), 301–303. MR 0010058, https://doi.org/10.1016/S0016-0032(44)90169-9
  • [65] G. N. Watson, R. So. London, Proc., 94A, 1918, p. 190-206.
  • [66] W. Koppe, Die Ausbreitung einer Erschütterung an den Wellenmaschinen, (Program no. 96, Andreas Realgymn. Berlin) Berlin, 1899, 28p. 1 table. See also J. W. Nicholson, Phil. Mag., s. 6, v. 16, 1908, p. 271, v. 18, 1909, p. 6.
  • [67] G. N. Watson, Bessel Functions, p. 248-252.
  • [68] V. A. Fok, loc. cit., no. 47.
  • [69] J. R. Airey, B.A.A.S., Report, 1922, p. 271-272, Jahnke & Emde 15, p. 143.
  • [70] W. N. Bailey, London Math. So., J., v. 4, 1929, p. 118-120.
  • [71] J. R. Airey, Phil. Mag., s. 6, v. 31, 1916, p. 520-528, v. 32, 1916, p. 7-14, v. 41, 1921, p. 200-205, B.A.A.S. Report, 1927, p. 252.
  • [72] Y. Ikeda, Zeit. Angew. Math. Mech., v. 5, 1925, p. 80-83.


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1944-0011175-4
Article copyright: © Copyright 1944 American Mathematical Society

American Mathematical Society