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1: 10.75 Tables
  • The main tables in Abramowitz and Stegun (1964, Chapter 9) give J 0 ( x ) to 15D, J 1 ( x ) , J 2 ( x ) , Y 0 ( x ) , Y 1 ( x ) to 10D, Y 2 ( x ) to 8D, x = 0 ( .1 ) 17.5 ; Y n ( x ) ( 2 / π ) J n ( x ) ln x , n = 0 , 1 , x = 0 ( .1 ) 2 , 8D; J n ( x ) , Y n ( x ) , n = 3 ( 1 ) 9 , x = 0 ( .2 ) 20 , 5D or 5S; J n ( x ) , Y n ( x ) , n = 0 ( 1 ) 20 ( 10 ) 50 , 100 , x = 1 , 2 , 5 , 10 , 50 , 100 , 10S; modulus and phase functions x M n ( x ) , θ n ( x ) x , n = 0 , 1 , 2 , 1 / x = 0 ( .01 ) 0.1 , 8D.

  • Young and Kirk (1964) tabulates ber n x , bei n x , ker n x , kei n x , n = 0 , 1 , x = 0 ( .1 ) 10 , 15D; ber n x , bei n x , ker n x , kei n x , modulus and phase functions M n ( x ) , θ n ( x ) , N n ( x ) , ϕ n ( x ) , n = 0 , 1 , 2 , x = 0 ( .01 ) 2.5 , 8S, and n = 0 ( 1 ) 10 , x = 0 ( .1 ) 10 , 7S. Also included are auxiliary functions to facilitate interpolation of the tables for n = 0 ( 1 ) 10 for small values of x . (Concerning the phase functions see §10.68(iv).)

  • Abramowitz and Stegun (1964, Chapter 9) tabulates ber n x , bei n x , ker n x , kei n x , n = 0 , 1 , x = 0 ( .1 ) 5 , 9–10D; x n ( ker n x + ( ber n x ) ( ln x ) ) , x n ( kei n x + ( bei n x ) ( ln x ) ) , n = 0 , 1 , x = 0 ( .1 ) 1 , 9D; modulus and phase functions M n ( x ) , θ n ( x ) , N n ( x ) , ϕ n ( x ) , n = 0 , 1 , x = 0 ( .2 ) 7 , 6D; x e x / 2 M n ( x ) , θ n ( x ) ( x / 2 ) , x e x / 2 N n ( x ) , ϕ n ( x ) + ( x / 2 ) , n = 0 , 1 , 1 / x = 0 ( .01 ) 0.15 , 5D.

  • Zhang and Jin (1996, p. 322) tabulates ber x , ber x , bei x , bei x , ker x , ker x , kei x , kei x , x = 0 ( 1 ) 20 , 7S.

  • Zhang and Jin (1996, p. 323) tabulates the first 20 real zeros of ber x , ber x , bei x , bei x , ker x , ker x , kei x , kei x , 8D.

  • 2: 10.73 Physical Applications
    See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). …
    §10.73(iii) Kelvin Functions
    The analysis of the current distribution in circular conductors leads to the Kelvin functions ber x , bei x , ker x , and kei x . …
    §10.73(iv) Bickley Functions
    3: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.
  • Y. Ameur and J. Cronvall (2023) Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • 4: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • K. Soni (1980) Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.