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Mathematics of Computation

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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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    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. 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. H. Weber, J.f. d. reine u. angew. Math., v. 76, 1873, p. 9; Math. Annalen, v. 6, 1873, p. 148. C. G. Neumann, Theorie der Bessel’schen Funktionen, Leipzig, 1867, p. 41. H. Hankel, Math. Annalen, v. 1, 1869, p. 471. 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. A. B. Basset, A Treatise on Hydrodynamics, with numerous Examples, 2v, Cambridge, 1881, v. 2, p. 15. J. Ivory, R. So. London, Trans., v. 113, 1823, p. 409, 495; and v. 128, 1838, part 2, p. 170-229. E. E. Kummer, J.f. d. reine u. angew. Math., v. 12, 1834, p. 144-147; and v. 17, 1837, p. 210-242. N. Nielsen, Handbuch der Theorie der Cylinderfunktionen, Leipzig, 1904, p. 16. P. Schafheitlin, Berlin Math. So., Sitzungsb., v. 8, 1909, p. 64. G. H. Hardy, “On certain definite integrals considered by Airy and Stokes,” Quart. J. Math., v. 41, 1910, p. 226-240. W. K. Clifford, Mathematical Papers, London, 1882, p. 346-349. G. Greenhill, Phil. Mag., s. 6, v. 38, 1919, p. 501-528. 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. E. Lommel, Math. Annalen, v. 9, 1876, p. 425-444. J. W. Nicholson, Quar. J. Math., v. 42, 1911, p. 216-224. N. Nielsen, Handbuch der Theorie der Cylinderfunktionen, Leipzig, 1904. L. Schläfli, Math. Annalen, v. 10, 1876, p. 137-142. 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. 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. 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. J. B. Costello, “Bessel product functions,” Phil. Mag., s. 7, v. 21, 1936, p. 308-318. A. Cauchy, Oeuvres, s. 1, v. 1, Paris, 1887, p. 277-278. 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. H. Poincaré, Acta Math., v. 8, 1886, p. 296 C. G. J. Jacobi, Ast. Nach., v. 28, 1848, col. 94; Werke, v. 7, p. 174. S. D. Poisson, École Polytechnique, Paris, J., v. 12, 1823, p. 349. A. Cauchy, Inst, de France, Acad. d. sci., Mémoires divers savans, v. 1, 1827, p. 272; Oeuvres, s. 1, v. 1, p. 277-278. E. W. Barnes, London Math. So., Proc., s. 2, v. 5, 1907, p. 59-116. J. Ivory, R. So. London, Trans., v. 113, 1823, p. 409-495, v. 128, 1838, p. 170-229. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge 1922, p. 207-210. N. S. Koshliakov, London Math. So., J., v. 4, 1929, p. 297-299. B. Strogonoff, Inst. Math. Stekloff, Travaux, v. 9, 1935, p. 223-233. 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. A. Lodge, B.A.A.S., Report, 1906, p. 494-498, 1907, p. 95-97. N. Nielsen, Handbuch der Theorie der Cylinderfunktionen, Leipzig, 1904. D. Burnett, Cambridge Phil. So., Proc., v. 26, 1930, p. 145-151. L. V. King, R. So. London, Trans., v. 214A, 1914, p. 373-432. K. Weierstrass, Akad. d. Wissen. Berlin, Sitzungsb. 1885, p. 633-639, 789-805, Werke v. 3, Berlin, 1903, p. 1-37. G. G. Stokes, Cambridge Phil. So. Trans., v. 10, 1857, p. 106-128, Math. & Phys. Papers, v. 4, p. 77-109.
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Article copyright: © Copyright 1944 American Mathematical Society