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1: 12.11 Zeros
§12.11(ii) Asymptotic Expansions of Large Zeros
12.11.5 p 0 ( ζ ) = t ( ζ ) ,
12.11.6 p 1 ( ζ ) = t 3 6 t 24 ( t 2 1 ) 2 + 5 48 ( ( t 2 1 ) ζ 3 ) 1 2 .
12.11.8 q 0 ( ζ ) = t ( ζ ) .
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 ) ,
2: 9.7 Asymptotic Expansions
Numerical values of χ ( n ) are given in Table 9.7.1 for n = 1 ( 1 ) 20 to 2D. …
§9.7(iii) Error Bounds for Real Variables
In (9.7.7) and (9.7.8) the n th error term is bounded in magnitude by the first neglected term multiplied by χ ( n + σ ) + 1 where σ = 1 6 for (9.7.7) and σ = 0 for (9.7.8), provided that n 0 in the first case and n 1 in the second case. …
§9.7(iv) Error Bounds for Complex Variables
provided that n 0 , σ = 1 6 for (9.7.5) and n 1 , σ = 0 for (9.7.6). …
3: 20.11 Generalizations and Analogs
In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
§20.11(iii) Ramanujan’s Change of Base
However, in this case q is no longer regarded as an independent complex variable within the unit circle, because k is related to the variable τ = τ ( k ) of the theta functions via (20.9.2). … These results are called Ramanujan’s changes of base. …
4: 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
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 ,
5.11.12 Γ ( z + a ) Γ ( z + b ) z a b ,
5.11.13 Γ ( z + a ) Γ ( z + b ) z a b k = 0 G k ( a , b ) z k ,
5: 12.10 Uniform Asymptotic Expansions for Large Parameter
In this section we give asymptotic expansions of PCFs for large values of the parameter a that are uniform with respect to the variable z , when both a and z ( = x ) are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables. … The variable ζ is defined by …
12.10.40 ϕ ( ζ ) = ( ζ t 2 1 ) 1 4 .
12.10.45 χ ( ζ ) = ϕ ( ζ ) ϕ ( ζ ) = 1 2 t ( ϕ ( ζ ) ) 6 4 ζ .
6: 19.36 Methods of Computation
Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). …If (19.36.1) is used instead of its first five terms, then the factor ( 3 r ) 1 / 6 in Carlson (1995, (2.2)) is changed to ( 3 r ) 1 / 8 . For both R D and R J the factor ( r / 4 ) 1 / 6 in Carlson (1995, (2.18)) is changed to ( r / 5 ) 1 / 8 when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: … Complex values of the variables are allowed, with some restrictions in the case of R J that are sufficient but not always necessary. … 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). …
7: Bibliography D
  • C. de la Vallée Poussin (1896a) Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction ζ ( s ) de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
  • C. de la Vallée Poussin (1896b) Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire M x + N . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.
  • 8: 20.10 Integrals