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1: 15.10 Hypergeometric Differential Equation
The ( 6 3 ) = 20 connection formulas for the principal branches of Kummer’s solutions are: …
2: 32.8 Rational Solutions
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 ,
32.8.9 w ( z ; n ) = d d z ( ln ( τ n 1 ( z ) τ n ( z ) ) ) ,
3: 6.16 Mathematical Applications
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
4: 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). 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
5: 27.2 Functions
Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …
27.2.3 π ( x ) x ln x .
27.2.4 p n n ln n .
27.2.14 Λ ( n ) = ln p , n = p a ,
Table 27.2.2: Functions related to division.
n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
7 6 2 8 20 8 6 42 33 20 4 48 46 22 4 72
6: 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 . …
5.11.8 Ln Γ ( z + h ) ( z + h 1 2 ) ln z z + 1 2 ln ( 2 π ) + k = 2 ( 1 ) k B k ( h ) k ( k 1 ) z k 1 ,
7: 10.75 Tables
  • Achenbach (1986) tabulates J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) , x = 0 ( .1 ) 8 , 20D or 18–20S.

  • Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of Y n ( z ) and Y n ( z ) for n = 0 ( 1 ) 5 , 8D.

  • Kerimov and Skorokhodov (1985b) tabulates 50 zeros of the principal branches of H 0 ( 1 ) ( z ) and H 1 ( 1 ) ( z ) , 8D.

  • 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.

  • 8: 9.9 Zeros
    9.9.6 a k = T ( 3 8 π ( 4 k 1 ) ) ,
    9.9.14 β k = e π i / 3 T ( 3 8 π ( 4 k 1 ) + 3 4 i ln 2 ) ,
    9.9.15 Bi ( β k ) = ( 1 ) k 2 e π i / 6 V ( 3 8 π ( 4 k 1 ) + 3 4 i ln 2 ) ,
    9.9.16 β k = e π i / 3 U ( 3 8 π ( 4 k 3 ) + 3 4 i ln 2 ) ,
    9.9.17 Bi ( β k ) = ( 1 ) k 1 2 e π i / 6 W ( 3 8 π ( 4 k 3 ) + 3 4 i ln 2 ) .
    9: 12.11 Zeros
    12.11.2 τ s = ( 2 s + 1 2 a ) π + i ln ( π 1 2 2 a 1 2 Γ ( 1 2 + a ) ) ,
    12.11.3 λ s = ln τ s 1 2 π i .
    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 ) ,
    10: 25.6 Integer Arguments
    25.6.3 ζ ( n ) = B n + 1 n + 1 , n = 1 , 2 , 3 , .
    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 ,
    25.6.15 ζ ( 2 n ) = ( 1 ) n + 1 ( 2 π ) 2 n 2 ( 2 n ) ! ( 2 n ζ ( 1 2 n ) ( ψ ( 2 n ) ln ( 2 π ) ) B 2 n ) .