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Bickley function

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1: 10.76 Approximations
Bickley Functions
2: 10.73 Physical Applications
§10.73(iv) Bickley Functions
3: 10.43 Integrals
§10.43(iii) Fractional Integrals
The Bickley function Ki α ( x ) is defined by …
10.43.14 Ki 0 ( x ) = K 0 ( x ) ,
For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos (1983c, 1989) and Luke (1962, Chapter 8).
§10.43(iv) Integrals over the Interval ( 0 , )
4: Bibliography
  • Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.
  • D. E. Amos (1983a) Algorithm 609. A portable FORTRAN subroutine for the Bickley functions Ki n ( x ) . ACM Trans. Math. Software 9 (4), pp. 480–493.
  • D. E. Amos (1983c) Uniform asymptotic expansions for exponential integrals E n ( x ) and Bickley functions Ki n ( x ) . ACM Trans. Math. Software 9 (4), pp. 467–479.
  • 5: Bibliography B
  • W. G. Bickley, L. J. Comrie, J. C. P. Miller, D. H. Sadler, and A. J. Thompson (1952) Bessel Functions. Part II: Functions of Positive Integer Order. British Association for the Advancement of Science, Mathematical Tables, Volume 10, Cambridge University Press, Cambridge.
  • 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.
  • J. M. Blair, C. A. Edwards, and J. H. Johnson (1978) Rational Chebyshev approximations for the Bickley functions K i n ( x ) . Math. Comp. 32 (143), pp. 876–886.
  • 6: 10.18 Modulus and Phase Functions
    §10.18 Modulus and Phase Functions
    §10.18(i) Definitions
    §10.18(ii) Basic Properties
    §10.18(iii) Asymptotic Expansions for Large Argument
    In (10.18.17) and (10.18.18) the remainder after n terms does not exceed the ( n + 1 ) th term in absolute value and is of the same sign, provided that n > ν 1 2 for (10.18.17) and 3 2 ν 3 2 for (10.18.18).
    7: 10.75 Tables
    §10.75 Tables
  • Bickley et al. (1952) tabulates J n ( x ) , Y n ( x ) or x n Y n ( x ) , n = 2 ( 1 ) 20 , x = 0 ( .01 or .1 ) 10 ( .1 ) 25 , 8D (for J n ( x ) ), 8S (for Y n ( x ) or x n Y n ( x ) ); J n ( x ) , Y n ( x ) , n = 0 ( 1 ) 20 , x = 0 or 0.1 ( .1 ) 25 , 10D (for J n ( x ) ), 10S (for Y n ( x ) ).

  • §10.75(iv) Integrals of Bessel Functions
  • 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.

  • Bickley and Nayler (1935) tabulates Ki n ( x ) 10.43(iii)) for n = 1 ( 1 ) 16 , x = 0 ( .05 ) 0.2 ( .1 ) 2, 3, 9D.

  • 8: 10.19 Asymptotic Expansions for Large Order
    §10.19 Asymptotic Expansions for Large Order
    §10.19(i) Asymptotic Forms
    §10.19(ii) Debye’s Expansions
    §10.19(iii) Transition Region
    See also §10.20(i).
    9: 10.21 Zeros
    For the next three terms in (10.21.19) and the next two terms in (10.21.20) see Bickley et al. (1952, p. xxxvii) or Olver (1960, pp. xvii–xviii). … The zeros of the functions
    §10.21(xiii) Rayleigh Function
    10: 10.41 Asymptotic Expansions for Large Order
    For U 4 ( p ) , U 5 ( p ) , U 6 ( p ) , see Bickley et al. (1952, p. xxxv). …
    §10.41(iv) Double Asymptotic Properties
    §10.41(v) Double Asymptotic Properties (Continued)