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1: 6.20 Approximations
  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • 2: 6.19 Tables
  • Zhang and Jin (1996, pp. 652, 689) includes Si ( x ) , Ci ( x ) , x = 0 ( .5 ) 20 ( 2 ) 30 , 8D; Ei ( x ) , E 1 ( x ) , x = [ 0 , 100 ] , 8S.

  • 3: Bibliography M
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • 4: 7.23 Tables
  • Abramowitz and Stegun (1964, Chapter 7) includes erf x , ( 2 / π ) e x 2 , x [ 0 , 2 ] , 10D; ( 2 / π ) e x 2 , x [ 2 , 10 ] , 8S; x e x 2 erfc x , x 2 [ 0 , 0.25 ] , 7D; 2 n Γ ( 1 2 n + 1 ) i n erfc ( x ) , n = 1 ( 1 ) 6 , 10 , 11 , x [ 0 , 5 ] , 6S; F ( x ) , x [ 0 , 2 ] , 10D; x F ( x ) , x 2 [ 0 , 0.25 ] , 9D; C ( x ) , S ( x ) , x [ 0 , 5 ] , 7D; f ( x ) , g ( x ) , x [ 0 , 1 ] , x 1 [ 0 , 1 ] , 15D.

  • Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral G ( x ) , x = 1 ( .1 ) 3 ( .5 ) 8 , 4D; also G ( x ) + ln x , x = 0 ( .05 ) 1 , 4D.

  • Zhang and Jin (1996, pp. 637, 639) includes ( 2 / π ) e x 2 , erf x , x = 0 ( .02 ) 1 ( .04 ) 3 , 8D; C ( x ) , S ( x ) , x = 0 ( .2 ) 10 ( 2 ) 100 ( 100 ) 500 , 8D.

  • 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.

  • Zhang and Jin (1996, p. 642) includes the first 10 zeros of erf z , 9D; the first 25 distinct zeros of C ( z ) and S ( z ) , 8S.

  • 5: 25.12 Polylogarithms
    The right-hand side is called Clausen’s integral. …
    Integral Representation
    §25.12(iii) Fermi–Dirac and Bose–Einstein Integrals
    The Fermi–Dirac and Bose–Einstein integrals are defined by … In terms of polylogarithms …
    6: 20.10 Integrals
    §20.10 Integrals
    20.10.1 0 x s 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 2 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
    20.10.4 0 e s t θ 1 ( β π 2 | i π t 2 ) d t = 0 e s t θ 2 ( ( 1 + β ) π 2 | i π t 2 ) d t = s sinh ( β s ) sech ( s ) ,
    20.10.5 0 e s t θ 3 ( ( 1 + β ) π 2 | i π t 2 ) d t = 0 e s t θ 4 ( β π 2 | i π t 2 ) d t = s cosh ( β s ) csch ( s ) .
    For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).
    7: 7.8 Inequalities
    7.8.5 x 2 2 x 2 + 1 x 2 ( 2 x 2 + 5 ) 4 x 4 + 12 x 2 + 3 x 𝖬 ( x ) < 2 x 4 + 9 x 2 + 4 4 x 4 + 20 x 2 + 15 < x 2 + 1 2 x 2 + 3 , x 0 .
    7.8.6 0 x e a t 2 d t < 1 3 a x ( 2 e a x 2 + a x 2 2 ) , a , x > 0 .
    7.8.7 sinh x 2 x < e x 2 F ( x ) = 0 x e t 2 d t < e x 2 1 x , x > 0 .
    The function F ( x ) / 1 e 2 x 2 is strictly decreasing for x > 0 . For these and similar results for Dawson’s integral F ( x ) see Janssen (2021). …
    8: 7.24 Approximations
    §7.24(i) Approximations in Terms of Elementary Functions
  • Cody (1969) provides minimax rational approximations for erf x and erfc x . The maximum relative precision is about 20S.

  • Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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

  • Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for F ( z ) , erf z , erfc z , C ( z ) , and S ( z ) ; approximate errors are given for a selection of z -values.

  • 9: 6.16 Mathematical Applications
    §6.16(i) The Gibbs Phenomenon
    Hence, if x is fixed and n , then S n ( x ) 1 4 π , 0 , or 1 4 π according as 0 < x < π , x = 0 , or π < x < 0 ; compare (6.2.14). … The first maximum of 1 2 Si ( x ) for positive x occurs at x = π and equals ( 1.1789 ) × 1 4 π ; compare Figure 6.3.2. …
    §6.16(ii) Number-Theoretic Significance of li ( x )
    If we assume Riemann’s hypothesis that all nonreal zeros of ζ ( s ) have real part of 1 2 25.10(i)), then …
    10: 36.4 Bifurcation Sets
    K = 1 , fold bifurcation set: …
    x = 9 20 z 2 .
    x = 3 20 z 2 ,
    y = 1 3 z 2 ( sin ( 2 ϕ ) 2 sin ϕ ) , 0 ϕ 2 π .