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1: 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)). … The principal branch has a cut along the interval [ 1 , ) and agrees with (25.12.1) when | z | 1 ; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches. …
See accompanying text
Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . Principal value. … Magnify 3D Help
For other values of z , Li s ( z ) is defined by analytic continuation. …
2: 10.3 Graphics
In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …
See accompanying text
Figure 10.3.10: H 0 ( 1 ) ( x + i y ) , 10 x 5 , 2.8 y 4 . Principal value. … Magnify 3D Help
See accompanying text
Figure 10.3.12: H 1 ( 1 ) ( x + i y ) , 10 x 5 , 2.8 y 4 . Principal value. … Magnify 3D Help
See accompanying text
Figure 10.3.14: H 5 ( 1 ) ( x + i y ) , 20 x 10 , 4 y 4 . Principal value. … Magnify 3D Help
See accompanying text
Figure 10.3.15: J 5.5 ( x + i y ) , 10 x 10 , 4 y 4 . Principal value. … Magnify 3D Help
3: 10.75 Tables
  • 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.

  • 4: 6.16 Mathematical Applications
    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. …
    6.16.5 li ( x ) π ( x ) = O ( x ln x ) , x ,
    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
    5: 5.11 Asymptotic Expansions
    5.11.1 Ln Γ ( z ) ( z 1 2 ) ln z z + 1 2 ln ( 2 π ) + k = 1 B 2 k 2 k ( 2 k 1 ) z 2 k 1
    5.11.2 ψ ( z ) ln z 1 2 z k = 1 B 2 k 2 k z 2 k .
    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 . … uniformly for bounded real values of x . …
    6: 32.8 Rational Solutions
    P II P VI  possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. …
    32.8.3 w ( z ; 3 ) = 3 z 2 z 3 + 4 6 z 2 ( z 3 + 10 ) z 6 + 20 z 3 80 ,
    32.8.4 w ( z ; 4 ) = 1 z + 6 z 2 ( z 3 + 10 ) z 6 + 20 z 3 80 9 z 5 ( z 3 + 40 ) z 9 + 60 z 6 + 11200 .
    32.8.5 w ( z ; n ) = d d z ( ln ( Q n 1 ( z ) Q n ( z ) ) ) ,
    Q 3 ( z ) = z 6 + 20 z 3 80 ,
    7: 20.7 Identities
    See Lawden (1989, pp. 19–20). … In the following equations τ = 1 / τ , and all square roots assume their principal values. …
    20.7.34 θ 1 ( z , q 2 ) θ 3 ( z , q 2 ) θ 1 ( z , i q ) = θ 2 ( z , q 2 ) θ 4 ( z , q 2 ) θ 2 ( z , i q ) = i 1 / 4 θ 2 ( 0 , q 2 ) θ 4 ( 0 , q 2 ) 2 .
    8: 19.36 Methods of Computation
    All cases of R F , R C , R J , and R D are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). 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). …
    9: 1.11 Zeros of Polynomials
    Resolvent cubic is z 3 + 12 z 2 + 20 z + 9 = 0 with roots θ 1 = 1 , θ 2 = 1 2 ( 11 + 85 ) , θ 3 = 1 2 ( 11 85 ) , and θ 1 = 1 , θ 2 = 1 2 ( 17 + 5 ) , θ 3 = 1 2 ( 17 5 ) . … where R = ( a 2 + b 2 ) 1 / 2 , α = ph ( a + i b ) , with the principal value of phase (§1.9(i)), and k = 0 , 1 , , n 1 . …
    10: 25.6 Integer Arguments
    §25.6(i) Function Values
    25.6.3 ζ ( n ) = B n + 1 n + 1 , n = 1 , 2 , 3 , .
    §25.6(ii) Derivative Values
    25.6.11 ζ ( 0 ) = 1 2 ln ( 2 π ) .
    25.6.12 ζ ′′ ( 0 ) = 1 2 ( ln ( 2 π ) ) 2 + 1 2 γ 2 1 24 π 2 + γ 1 ,