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Abel–Plana formula

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11: 5.5 Functional Relations
§5.5(ii) Reflection
5.5.3 Γ ( z ) Γ ( 1 - z ) = π / sin ( π z ) , z 0 , ± 1 , ,
§5.5(iii) Multiplication
Duplication Formula
Gauss’s Multiplication Formula
12: 24.6 Explicit Formulas
§24.6 Explicit Formulas
24.6.6 E 2 n = k = 1 2 n ( - 1 ) k 2 k - 1 ( 2 n + 1 k + 1 ) j = 0 1 2 k - 1 2 ( k j ) ( k - 2 j ) 2 n .
24.6.7 B n ( x ) = k = 0 n 1 k + 1 j = 0 k ( - 1 ) j ( k j ) ( x + j ) n ,
24.6.12 E 2 n = k = 0 2 n 1 2 k j = 0 k ( - 1 ) j ( k j ) ( 1 + 2 j ) 2 n .
13: 27.5 Inversion Formulas
§27.5 Inversion Formulas
which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … Special cases of Möbius inversion pairs are: … Other types of Möbius inversion formulas include: …
14: 3.5 Quadrature
Gauss–Legendre Formula
Gauss–Chebyshev Formula
Gauss–Laguerre Formula
a complex Gauss quadrature formula is available. …
15: Possible Errors in DLMF
One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …
16: 18.42 Software
A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index. …
17: 25.4 Reflection Formulas
§25.4 Reflection Formulas
25.4.1 ζ ( 1 - s ) = 2 ( 2 π ) - s cos ( 1 2 π s ) Γ ( s ) ζ ( s ) ,
25.4.2 ζ ( s ) = 2 ( 2 π ) s - 1 sin ( 1 2 π s ) Γ ( 1 - s ) ζ ( 1 - s ) .
25.4.3 ξ ( s ) = ξ ( 1 - s ) ,
18: 2.2 Transcendental Equations
where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …
19: 25.19 Tables
  • Cloutman (1989) tabulates Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for s = - 1 2 , 1 2 , 3 2 , 5 2 , x = - 5 ( .05 ) 25 , to 12S.

  • Fletcher et al. (1962, §22.1) lists many sources for earlier tables of ζ ( s ) for both real and complex s . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of ζ ( s , a ) , and §22.17 lists tables for some Dirichlet L -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

  • 20: 29.20 Methods of Computation
    Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … §29.15(i) includes formulas for normalizing the eigenvectors. …