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1: 6.16 Mathematical Applications
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. Hence if x = π / ( 2 n ) and n , then the limiting value of S n ( x ) overshoots 1 4 π by approximately 18%. Similarly if x = π / n , then the limiting value of S n ( x ) undershoots 1 4 π by approximately 10%, and so on. …
See accompanying text
Figure 6.16.2: The logarithmic integral li ( x ) , together with vertical bars indicating the value of π ( x ) for x = 10 , 20 , , 1000 . Magnify
2: 19.36 Methods of Computation
Complex values of the variables are allowed, with some restrictions in the case of R J that are sufficient but not always necessary. … Accurate values of F ( ϕ , k ) E ( ϕ , k ) for k 2 near 0 can be obtained from R D by (19.2.6) and (19.25.13). … This method loses significant figures in ρ if α 2 and k 2 are nearly equal unless they are given exact values—as they can be for tables. … For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … When the values of complete integrals are known, addition theorems with ψ = π / 2 19.11(ii)) ease the computation of functions such as F ( ϕ , k ) when 1 2 π ϕ is small and positive. …
3: 10.75 Tables
  • Olver (1960) tabulates j n , m , J n ( j n , m ) , j n , m , J n ( j n , m ) , y n , m , Y n ( y n , m ) , y n , m , Y n ( y n , m ) , n = 0 ( 1 2 ) 20 1 2 , m = 1 ( 1 ) 50 , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n ; see §10.21(viii), and more fully Olver (1954).

  • Makinouchi (1966) tabulates all values of j ν , m and y ν , m in the interval ( 0 , 100 ) , with at least 29S. These are for ν = 0 ( 1 ) 5 , 10, 20; ν = 3 2 , 5 2 ; ν = m / n with m = 1 ( 1 ) n 1 and n = 3 ( 1 ) 8 , except for ν = 1 2 .

  • Zhang and Jin (1996, p. 270) tabulates 0 x J 0 ( t ) d t , 0 x t 1 ( 1 J 0 ( t ) ) d t , 0 x Y 0 ( t ) d t , x t 1 Y 0 ( t ) d t , x = 0 ( .1 ) 1 ( .5 ) 20 , 8D.

  • Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of K n ( z ) and K n ( z ) , for n = 2 ( 1 ) 20 , 9S.

  • Olver (1960) tabulates a n , m , 𝗃 n ( a n , m ) , b n , m , 𝗒 n ( b n , m ) , n = 1 ( 1 ) 20 , m = 1 ( 1 ) 50 , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n .

  • 4: 5.11 Asymptotic Expansions
    Wrench (1968) gives exact values of g k up to g 20 . Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of g k for k = 21 , 22 , , 30 . … Lastly, as y ± , …uniformly for bounded real values of x . … If the sums in the expansions (5.11.1) and (5.11.2) are terminated at k = n 1 ( k 0 ) and z is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. …
    5: 26.10 Integer Partitions: Other Restrictions
    p m ( 𝒟 , n ) denotes the number of partitions of n into at most m distinct parts. p ( 𝒟 k , n ) denotes the number of partitions of n into parts with difference at least k . p ( 𝒟 3 , n ) denotes the number of partitions of n into parts with difference at least 3, except that multiples of 3 must differ by at least 6. … where the sum is over nonnegative integer values of k for which n 1 2 ( 3 k 2 ± k ) 0 . … The quantity A k ( n ) is real-valued. …
    6: 2.11 Remainder Terms; Stokes Phenomenon
    If the results agree within S significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least S correct significant figures for larger values of x . … In this way we arrive at hyperasymptotic expansions. … 17408, compared with the correct valueComparison with the true valueFor example, using double precision d 20 is found to agree with (2.11.31) to 13D. …
    7: 12.11 Zeros
    If a > 1 2 , then V ( a , x ) has no positive real zeros, and if a = 3 2 2 n , n , then V ( a , x ) has a zero at x = 0 . … When a > 1 2 , U ( a , z ) has a string of complex zeros that approaches the ray ph z = 3 4 π as z , and a conjugate string. … For large negative values of a the real zeros of U ( a , x ) , U ( a , x ) , V ( a , x ) , and V ( a , x ) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …as μ ( = 2 a ) , s fixed. …
    12.11.9 u a , 1 2 1 2 μ ( 1 1.85575 708 μ 4 / 3 0.34438 34 μ 8 / 3 0.16871 5 μ 4 0.11414 μ 16 / 3 0.0808 μ 20 / 3 ) ,
    8: 25.12 Polylogarithms
    Other notations and names for Li 2 ( z ) include S 2 ( z ) (Kölbig et al. (1970)), Spence function Sp ( z ) (’t Hooft and Veltman (1979)), and L 2 ( z ) (Maximon (2003)). In the complex plane Li 2 ( z ) has a branch point at z = 1 . The principal branch has a cut along the interval [ 1 , ) and agrees with (25.12.1) when | z | 1 ; see also §4.2(i). …
    See accompanying text
    Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . … Magnify 3D Help
    For other values of z , Li s ( z ) is defined by analytic continuation. …
    9: 25.11 Hurwitz Zeta Function
    ζ ( s , a ) has a meromorphic continuation in the s -plane, its only singularity in being a simple pole at s = 1 with residue 1 . …
    §25.11(v) Special Values
    As β ± with s fixed, s > 1 , …uniformly with respect to bounded nonnegative values of α . As a in the sector | ph a | π δ ( < π ) , with s ( 1 ) and δ fixed, we have the asymptotic expansion …
    10: Errata
  • Section 11.11

    The asymptotic results were originally for ν real valued and ν + . However, these results are also valid for complex values of ν . The maximum sectors of validity are now specified.

  • Subsection 14.2(iii)

    Previously the exponents of the associated Legendre differential equation (14.2.2) at infinity were given incorrectly by { ν 1 , ν } . These were replaced by { ν + 1 , ν } .

    Reported by Hans Volkmer on 2019-01-30

  • Subsection 13.8(iii)

    A new paragraph with several new equations and a new reference has been added at the end to provide asymptotic expansions for Kummer functions U ( a , b , z ) and 𝐌 ( a , b , z ) as a in | ph a | π δ and b and z fixed.

  • Chapters 8, 20, 36

    Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

  • References

    Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.