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1: 10.73 Physical Applications
Bessel functions of the first kind, J n ( x ) , arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation 2 V = 0 , or by the Helmholtz equation ( 2 + k 2 ) ψ = 0 . … In cylindrical coordinates r , ϕ , z , (§1.5(ii) we have … Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces. … On separation of variables into cylindrical coordinates, the Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) and K n ( x ) , all appear. …
2: Bibliography K
  • M. K. Kerimov and S. L. Skorokhodov (1986) On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).
  • M. K. Kerimov and S. L. Skorokhodov (1987) On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.
  • M. K. Kerimov and S. L. Skorokhodov (1988) Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).
  • M. Kodama (2008) Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.
  • M. Kodama (2011) Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument. ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.
  • 3: 1.5 Calculus of Two or More Variables
    Cylindrical Coordinates
    1.5.15 2 f = 2 f x 2 + 2 f y 2 + 2 f z 2 = 2 f r 2 + 1 r f r + 1 r 2 2 f ϕ 2 + 2 f z 2 .
    4: Bibliography H
  • P. I. Hadži (1972) Certain sums that contain cylindrical functions. Bul. Akad. Štiince RSS Moldoven. 1972 (3), pp. 75–77, 94 (Russian).
  • P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
  • 5: Bibliography T
  • C. A. Tracy and H. Widom (1997) On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.
  • 6: Bibliography S
  • K. M. Siegel and F. B. Sleator (1954) Inequalities involving cylindrical functions of nearly equal argument and order. Proc. Amer. Math. Soc. 5 (3), pp. 337–344.
  • R. Spigler (1980) Some results on the zeros of cylindrical functions and of their derivatives. Rend. Sem. Mat. Univ. Politec. Torino 38 (1), pp. 67–85 (Italian. English summary).
  • 7: Bibliography
  • M. M. Agrest and M. S. Maksimov (1971) Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.
  • 8: Bibliography L
  • D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.
  • 9: Bibliography C
  • C. J. Chapman (1999) Caustics in cylindrical ducts. Proc. Roy. Soc. London Ser. A 455, pp. 2529–2548.
  • 10: Bibliography M
  • V. P. Modenov and A. V. Filonov (1986) Calculation of zeros of cylindrical functions and their derivatives. Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. (2), pp. 63–64, 71 (Russian).