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Glaisher constant

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1: 5.17 Barnes’ G -Function (Double Gamma Function)
5.17.5 Ln G ( z + 1 ) 1 4 z 2 + z Ln Γ ( z + 1 ) - ( 1 2 z ( z + 1 ) + 1 12 ) Ln z - ln A + k = 1 B 2 k + 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) z 2 k .
Here B 2 k + 2 is the Bernoulli number (§24.2(i)), and A is Glaisher’s constant, given by
5.17.6 A = e C = 1.28242 71291 00622 63687 ,
5.17.7 C = lim n ( k = 1 n k ln k - ( 1 2 n 2 + 1 2 n + 1 12 ) ln n + 1 4 n 2 ) = γ + ln ( 2 π ) 12 - ζ ( 2 ) 2 π 2 = 1 12 - ζ ( - 1 ) ,
For Glaisher’s constant see also Greene and Knuth (1982, p. 100) and §2.10(i).
2: 22.1 Special Notation
The notation sn ( z , k ) , cn ( z , k ) , dn ( z , k ) is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for sn ( z , k ) are sn ( z | m ) and sn ( z , m ) with m = k 2 ; see Abramowitz and Stegun (1964) and Walker (1996). …
3: 27.21 Tables
Glaisher (1940) contains four tables: Table I tabulates, for all n 10 4 : (a) the canonical factorization of n into powers of primes; (b) the Euler totient ϕ ( n ) ; (c) the divisor function d ( n ) ; (d) the sum σ ( n ) of these divisors. …7 of Abramowitz and Stegun (1964) also lists the factorizations in Glaisher’s Table I(a); Table 24. …
4: 2.10 Sums and Sequences
Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). …where γ is Euler’s constant5.2(ii)) and ζ is the derivative of the Riemann zeta function (§25.2(i)). e C is sometimes called Glaisher’s constant. … where α ( - 1 ) is a real constant, and … for any real constant α and the set of all positive integers j , we derive …
5: 22.4 Periods, Poles, and Zeros
§22.4(ii) Graphical Interpretation via Glaisher’s Notation
6: 3.12 Mathematical Constants
§3.12 Mathematical Constants
The fundamental constant …Other constants that appear in the DLMF include the base e of natural logarithms …see §4.2(ii), and Euler’s constant γ For access to online high-precision numerical values of mathematical constants see Sloane (2003). …
7: 30.1 Special Notation
x real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, - 1 < x < 1 .
δ arbitrary small positive constant.
The main functions treated in this chapter are the eigenvalues λ n m ( γ 2 ) and the spheroidal wave functions Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) , Ps n m ( z , γ 2 ) , Qs n m ( z , γ 2 ) , and S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 . …Meixner and Schäfke (1954) use ps , qs , Ps , Qs for Ps , Qs , Ps , Qs , respectively. … Flammer (1957) and Abramowitz and Stegun (1964) use λ m n ( γ ) for λ n m ( γ 2 ) + γ 2 , R m n ( j ) ( γ , z ) for S n m ( j ) ( z , γ ) , and …where d m n ( γ ) is a normalization constant determined by …
8: 16.15 Integral Representations and Integrals
16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ - α ) 0 1 u α - 1 ( 1 - u ) γ - α - 1 ( 1 - u x ) β ( 1 - u y ) β d u , α > 0 , ( γ - α ) > 0 ,
16.15.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β ) Γ ( γ - β ) 0 1 0 1 u β - 1 v β - 1 ( 1 - u ) γ - β - 1 ( 1 - v ) γ - β - 1 ( 1 - u x - v y ) α d u d v , γ > β > 0 , γ > β > 0 ,
16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
16.15.4 F 4 ( α , β ; γ , γ ; x ( 1 - y ) , y ( 1 - x ) ) = Γ ( γ ) Γ ( γ ) Γ ( α ) Γ ( β ) Γ ( γ - α ) Γ ( γ - β ) 0 1 0 1 u α - 1 v β - 1 ( 1 - u ) γ - α - 1 ( 1 - v ) γ - β - 1 ( 1 - u x ) γ + γ - α - 1 ( 1 - v y ) γ + γ - β - 1 ( 1 - u x - v y ) α + β - γ - γ + 1 d u d v , γ > α > 0 , γ > β > 0 .
9: 32.9 Other Elementary Solutions
with κ , λ , μ , and ν arbitrary constants. … with C an arbitrary constant, which is solvable by quadrature. … with κ and μ arbitrary constants. … with C an arbitrary constant, which is solvable by quadrature. … with κ and μ arbitrary constants. …
10: 30.5 Functions of the Second Kind
Other solutions of (30.2.1) with μ = m , λ = λ n m ( γ 2 ) , and z = x are
30.5.1 Qs n m ( x , γ 2 ) , n = m , m + 1 , m + 2 , .
30.5.2 Qs n m ( - x , γ 2 ) = ( - 1 ) n - m + 1 Qs n m ( x , γ 2 ) ,
30.5.4 𝒲 { Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) } = ( n + m ) ! ( 1 - x 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ( 0 ) ,
with A n ± m ( γ 2 ) as in (30.11.4). …