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1: 26.10 Integer Partitions: Other Restrictions
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 . … where the sum is over nonnegative integer values of k for which n ( 3 k 2 ± k ) 0 . … where the sum is over nonnegative integer values of m for which n 1 2 k m 2 m + 1 2 k m 0 . … The quantity A k ( n ) is real-valued. …
2: 34.2 Definition: 3 j Symbol
§34.2 Definition: 3 j Symbol
The quantities j 1 , j 2 , j 3 in the 3 j symbol are called angular momenta. …They therefore satisfy the triangle conditions …where r , s , t is any permutation of 1 , 2 , 3 . The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by …
3: 10.75 Tables
  • 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 .

  • 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 .

  • 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.

  • 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: 6.16 Mathematical Applications
    6.16.1 sin x + 1 3 sin ( 3 x ) + 1 5 sin ( 5 x ) + = { 1 4 π , 0 < x < π , 0 , x = 0 , 1 4 π , π < x < 0 .
    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
    5: 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. …
    6: 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 . …
    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. …
    7: 28.16 Asymptotic Expansions for Large q
    §28.16 Asymptotic Expansions for Large q
    28.16.1 λ ν ( h 2 ) 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
    8: 27.20 Methods of Computation: Other Number-Theoretic Functions
    To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). … To compute a particular value p ( n ) it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
    9: 20 Theta Functions
    Chapter 20 Theta Functions
    10: 3.8 Nonlinear Equations
    From this graph we estimate an initial value x 0 = 4.65 . … The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of q ( z ) . … Table 3.8.3 gives the successive values of s and t . … Consider x = 20 and j = 19 . We have p ( 20 ) = 19 ! and a 19 = 1 + 2 + + 20 = 210 . …