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1: 24.20 Tables
§24.20 Tables
Abramowitz and Stegun (1964, Chapter 23) includes exact values of k = 1 m k n , m = 1 ( 1 ) 100 , n = 1 ( 1 ) 10 ; k = 1 k n , k = 1 ( 1 ) k 1 k n , k = 0 ( 2 k + 1 ) n , n = 1 , 2 , , 20D; k = 0 ( 1 ) k ( 2 k + 1 ) n , n = 1 , 2 , , 18D. Wagstaff (1978) gives complete prime factorizations of N n and E n for n = 20 ( 2 ) 60 and n = 8 ( 2 ) 42 , respectively. …
2: 27.2 Functions
Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …It can be expressed as a sum over all primes p x : … This is the number of positive integers n that are relatively prime to n ; ϕ ( n ) is Euler’s totient. If ( a , n ) = 1 , then the Euler–Fermat theorem states that … Table 27.2.2 tabulates the Euler totient function ϕ ( n ) , the divisor function d ( n ) ( = σ 0 ( n ) ), and the sum of the divisors σ ( n ) ( = σ 1 ( n ) ), for n = 1 ( 1 ) 52 . …
3: 24.2 Definitions and Generating Functions
§24.2(ii) Euler Numbers and Polynomials
24.2.10 E n ( x ) = k = 0 n ( n k ) E k 2 k ( x 1 2 ) n k .
§24.2(iii) Periodic Bernoulli and Euler Functions
Table 24.2.1: Bernoulli and Euler numbers.
n B n E n
Table 24.2.6: Coefficients e n , k of the Euler polynomials E n ( x ) = k = 0 n e n , k x k .
k
4: 11.6 Asymptotic Expansions
11.6.1 𝐊 ν ( z ) 1 π k = 0 Γ ( k + 1 2 ) ( 1 2 z ) ν 2 k 1 Γ ( ν + 1 2 k ) , | ph z | π δ ,
11.6.2 𝐌 ν ( z ) 1 π k = 0 ( 1 ) k + 1 Γ ( k + 1 2 ) ( 1 2 z ) ν 2 k 1 Γ ( ν + 1 2 k ) , | ph z | 1 2 π δ .
11.6.3 0 z 𝐊 0 ( t ) d t 2 π ( ln ( 2 z ) + γ ) 2 π k = 1 ( 1 ) k + 1 ( 2 k ) ! ( 2 k 1 ) ! ( k ! ) 2 ( 2 z ) 2 k , | ph z | π δ ,
where γ is Euler’s constant (§5.2(ii)). …
c 3 ( λ ) = 20 λ 6 4 λ 4 ,
5: 25.12 Polylogarithms
The notation Li 2 ( z ) was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … The cosine series in (25.12.7) has the elementary sum
See accompanying text
Figure 25.12.1: Dilogarithm function Li 2 ( x ) , 20 x < 1 . Magnify
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
Sometimes the factor 1 / Γ ( s + 1 ) is omitted. …
6: 25.11 Hurwitz Zeta Function
See accompanying text
Figure 25.11.1: Hurwitz zeta function ζ ( x , a ) , a = 0. …8, 1, 20 x 10 . … Magnify
§25.11(iii) Representations by the Euler–Maclaurin Formula
25.11.10 ζ ( s , a ) = n = 0 ( s ) n n ! ζ ( n + s ) ( 1 a ) n , s 1 , | a 1 | < 1 .
§25.11(xi) Sums
For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360). …
7: 5.11 Asymptotic Expansions
The scaled gamma function Γ ( z ) is defined in (5.11.3) and its main property is Γ ( z ) 1 as z in the sector | ph z | π δ . Wrench (1968) gives exact values of g k up to g 20 . … 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. …
5.11.13 Γ ( z + a ) Γ ( z + b ) z a b k = 0 G k ( a , b ) z k ,
5.11.19 Γ ( z + a ) Γ ( z + b ) Γ ( z + c ) k = 0 ( 1 ) k ( c a ) k ( c b ) k k ! Γ ( a + b c + z k ) .
8: 25.6 Integer Arguments
§25.6(i) Function Values
25.6.3 ζ ( n ) = B n + 1 n + 1 , n = 1 , 2 , 3 , .
where γ 1 is given by (25.2.5). …
25.6.13 ( 1 ) k ζ ( k ) ( 2 n ) = 2 ( 1 ) n ( 2 π ) 2 n + 1 m = 0 k r = 0 m ( k m ) ( m r ) ( c k m ) Γ ( r ) ( 2 n + 1 ) ζ ( m r ) ( 2 n + 1 ) ,
25.6.14 ( 1 ) k ζ ( k ) ( 1 2 n ) = 2 ( 1 ) n ( 2 π ) 2 n m = 0 k r = 0 m ( k m ) ( m r ) ( c k m ) Γ ( r ) ( 2 n ) ζ ( m r ) ( 2 n ) ,
9: 8.17 Incomplete Beta Functions
where, as in §5.12, B ( a , b ) denotes the beta function:
8.17.3 B ( a , b ) = Γ ( a ) Γ ( b ) Γ ( a + b ) .
§8.17(vi) Sums
For sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62). …
8.17.24 I x ( m , n ) = ( 1 x ) n j = m ( n + j 1 j ) x j , m , n positive integers; 0 x < 1 .
10: 9.7 Asymptotic Expansions
9.7.3 χ ( x ) π 1 / 2 Γ ( 1 2 x + 1 ) / Γ ( 1 2 x + 1 2 ) .
Numerical values of χ ( n ) are given in Table 9.7.1 for n = 1 ( 1 ) 20 to 2D. …
9.7.20 R n ( z ) = ( 1 ) n k = 0 m 1 ( 1 ) k u k G n k ( 2 ζ ) ζ k + R m , n ( z ) ,
9.7.21 S n ( z ) = ( 1 ) n 1 k = 0 m 1 ( 1 ) k v k G n k ( 2 ζ ) ζ k + S m , n ( z ) ,
9.7.22 G p ( z ) = e z 2 π Γ ( p ) Γ ( 1 p , z ) .