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1: 6.20 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for E 1 ( x ) + ln x , x e x E 1 ( x ) , and the auxiliary functions f ( x ) and g ( x ) . These are included in Abramowitz and Stegun (1964, Ch. 5).

  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein ( a x ) , Si ( a x ) , and Cin ( a x ) for 1 x 1 , a . The coefficients are given in terms of series of Bessel functions.

  • 2: 22.3 Graphics
    §22.3(i) Real Variables: Line Graphs
    §22.3(iii) Complex z ; Real k
    In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …
    §22.3(iv) Complex k
    In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase. …
    3: 7.23 Tables
  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf z , x [ 0 , 5 ] , y = 0.5 ( .5 ) 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i t 2 d t , ( 1 / π ) e i ( x 2 + ( π / 4 ) ) x e ± i t 2 d t , x = 0 ( .5 ) 20 ( 1 ) 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

  • 4: 10.75 Tables
  • 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 .

  • 5: 36.2 Catastrophes and Canonical Integrals
    36.2.28 Ψ ( E ) ( 0 , 0 , z ) = Ψ ( E ) ( 0 , 0 , z ) ¯ = 2 π π z 27 exp ( 2 27 i z 3 ) ( J 1 / 6 ( 2 27 z 3 ) + i J 1 / 6 ( 2 27 z 3 ) ) , z 0 ,
    36.2.29 Ψ ( H ) ( 0 , 0 , z ) = Ψ ( H ) ( 0 , 0 , z ) ¯ = 2 1 / 3 3 exp ( 1 27 i z 3 ) Ψ ( E ) ( 0 , 0 , z 2 2 / 3 ) , < z < .
    6: 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.
  • 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 (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.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • 7: 7.24 Approximations
  • Cody (1969) provides minimax rational approximations for erf x and erfc x . The maximum relative precision is about 20S.

  • Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • Luke (1969b, pp. 323–324) covers 1 2 π erf x and e x 2 F ( x ) for 3 x 3 (the Chebyshev coefficients are given to 20D); π x e x 2 erfc x and 2 x F ( x ) for x 3 (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

  • Schonfelder (1978) gives coefficients of Chebyshev expansions for x 1 erf x on 0 x 2 , for x e x 2 erfc x on [ 2 , ) , and for e x 2 erfc x on [ 0 , ) (30D).

  • Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for ( 1 + 2 x ) e x 2 erfc x on ( 0 , ) (22D).

  • 8: 9.18 Tables
  • Miller (1946) tabulates Ai ( x ) , Ai ( x ) for x = 20 ( .01 ) 2 ; log 10 Ai ( x ) , Ai ( x ) / Ai ( x ) for x = 0 ( .1 ) 25 ( 1 ) 75 ; Bi ( x ) , Bi ( x ) for x = 10 ( .1 ) 2.5 ; log 10 Bi ( x ) , Bi ( x ) / Bi ( x ) for x = 0 ( .1 ) 10 ; M ( x ) , N ( x ) , θ ( x ) , ϕ ( x ) (respectively F ( x ) , G ( x ) , χ ( x ) , ψ ( x ) ) for x = 80 ( 1 ) 30 ( .1 ) 0 . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

  • Fox (1960, Table 3) tabulates 2 π 1 / 2 x 1 / 4 exp ( 2 3 x 3 / 2 ) Ai ( x ) , 2 π 1 / 2 x 1 / 4 exp ( 2 3 x 3 / 2 ) Ai ( x ) , π 1 / 2 x 1 / 4 exp ( 2 3 x 3 / 2 ) Bi ( x ) , and π 1 / 2 x 1 / 4 exp ( 2 3 x 3 / 2 ) Bi ( x ) for 3 2 x 3 / 2 = 0 ( .001 ) 0.05 , together with similar auxiliary functions for negative values of x . Precision is 10D.

  • Zhang and Jin (1996, p. 337) tabulates Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) for x = 0 ( 1 ) 20 to 8S and for x = 20 ( 1 ) 0 to 9D.

  • Miller (1946) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; b k , Bi ( b k ) , b k , Bi ( b k ) , k = 1 ( 1 ) 20 . Precision is 8D. Entries for k = 1 ( 1 ) 20 are reproduced in Abramowitz and Stegun (1964, Chapter 10).

  • Sherry (1959) tabulates a k , Ai ( a k ) , a k , Ai ( a k ) , k = 1 ( 1 ) 50 ; 20S.

  • 9: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • §8.26(iv) Generalized Exponential Integral
  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 10: 14.30 Spherical and Spheroidal Harmonics
    §14.30 Spherical and Spheroidal Harmonics
    Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics R n m ( x ) = e i π n / 2 P n m ( i x ) and T n m ( x ) = i e i π n / 2 Q n m ( i x ) which are real when x > 0 and n = 0 , 1 , 2 , . …
    Herglotz generating function
    The following is the Herglotz generating function