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1: 33.24 Tables
§33.24 Tables
  • Abramowitz and Stegun (1964, Chapter 14) tabulates F 0 ( η , ρ ) , G 0 ( η , ρ ) , F 0 ( η , ρ ) , and G 0 ( η , ρ ) for η = 0.5 ( .5 ) 20 and ρ = 1 ( 1 ) 20 , 5S; C 0 ( η ) for η = 0 ( .05 ) 3 , 6S.

  • Curtis (1964a) tabulates P ( ϵ , r ) , Q ( ϵ , r ) 33.1), and related functions for = 0 , 1 , 2 and ϵ = 2 ( .2 ) 2 , with x = 0 ( .1 ) 4 for ϵ < 0 and x = 0 ( .1 ) 10 for ϵ 0 ; 6D.

  • 2: 33.3 Graphics
    §33.3 Graphics
    §33.3(i) Line Graphs of the Coulomb Radial Functions F ( η , ρ ) and G ( η , ρ )
    33.3.1 M ( η , ρ ) = ( F 2 ( η , ρ ) + G 2 ( η , ρ ) ) 1 / 2 = | H ± ( η , ρ ) | .
    §33.3(ii) Surfaces of the Coulomb Radial Functions F 0 ( η , ρ ) and G 0 ( η , ρ )
    See accompanying text
    Figure 33.3.8: G 0 ( η , ρ ) , 2 η 2 , 0 < ρ 5 . Magnify 3D Help
    3: 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.

  • Achenbach (1986) tabulates J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) , x = 0 ( .1 ) 8 , 20D or 18–20S.

  • British Association for the Advancement of Science (1937) tabulates I 0 ( x ) , I 1 ( x ) , x = 0 ( .001 ) 5 , 7–8D; K 0 ( x ) , K 1 ( x ) , x = 0.01 ( .01 ) 5 , 7–10D; e x I 0 ( x ) , e x I 1 ( x ) , e x K 0 ( x ) , e x K 1 ( x ) , x = 5 ( .01 ) 10 ( .1 ) 20 , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of K 0 ( x ) , K 1 ( x ) for small values of x .

  • Bickley et al. (1952) tabulates x n I n ( x ) or e x I n ( x ) , x n K n ( x ) or e x K n ( x ) , n = 2 ( 1 ) 20 , x = 0 (.01 or .1) 10(.1) 20, 8S; I n ( x ) , K n ( x ) , n = 0 ( 1 ) 20 , x = 0 or 0.1 ( .1 ) 20 , 10S.

  • Zhang and Jin (1996, pp. 296–305) tabulates 𝗃 n ( x ) , 𝗃 n ( x ) , 𝗒 n ( x ) , 𝗒 n ( x ) , 𝗂 n ( 1 ) ( x ) , 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , 𝗄 n ( x ) , n = 0 ( 1 ) 10 ( 10 ) 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; x 𝗃 n ( x ) , ( x 𝗃 n ( x ) ) , x 𝗒 n ( x ) , ( x 𝗒 n ( x ) ) (Riccati–Bessel functions and their derivatives), n = 0 ( 1 ) 10 ( 10 ) 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; real and imaginary parts of 𝗃 n ( z ) , 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗒 n ( z ) , 𝗂 n ( 1 ) ( z ) , 𝗂 n ( 1 ) ( z ) , 𝗄 n ( z ) , 𝗄 n ( z ) , n = 0 ( 1 ) 15 , 20(10)50, 100, z = 4 + 2 i , 20 + 10 i , 8S. (For the notation replace j , y , i , k by 𝗃 , 𝗒 , 𝗂 ( 1 ) , 𝗄 , respectively.)

  • 4: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • T. D. Newton (1952) Coulomb Functions for Large Values of the Parameter η . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • N. E. Nørlund (1955) Hypergeometric functions. Acta Math. 94, pp. 289–349.
  • 5: Bibliography M
  • I. G. Macdonald (1972) Affine root systems and Dedekind’s η -function. Invent. Math. 15 (2), pp. 91–143.
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • N. Michel (2007) Precise Coulomb wave functions for a wide range of complex , η and z . Computer Physics Communications 176 (3), pp. 232–249.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • 6: Bibliography B
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • C. Bardin, Y. Dandeu, L. Gauthier, J. Guillermin, T. Lena, J. M. Pernet, H. H. Wolter, and T. Tamura (1972) Coulomb functions in entire ( η , ρ )-plane. Comput. Phys. Comm. 3 (2), pp. 73–87.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley and J. Nayler (1935) A short table of the functions Ki n ( x ) , from n = 1 to n = 16 . Phil. Mag. Series 7 20, pp. 343–347.
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • 7: Bibliography D
  • C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire M x + N . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
  • R. B. Dingle (1957a) The Bose-Einstein integrals p ( η ) = ( p ! ) 1 0 ϵ p ( e ϵ η 1 ) 1 𝑑 ϵ . Appl. Sci. Res. B. 6, pp. 240–244.
  • R. B. Dingle (1957b) The Fermi-Dirac integrals p ( η ) = ( p ! ) 1 0 ϵ p ( e ϵ η + 1 ) 1 𝑑 ϵ . Appl. Sci. Res. B. 6, pp. 225–239.
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • 8: 12.10 Uniform Asymptotic Expansions for Large Parameter
    The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions2.8(iii)). …
    §12.10(vi) Modifications of Expansions in Elementary Functions
    where ξ , η are given by (12.10.7), (12.10.23), respectively, and …
    Modified Expansions