About the Project

Abel–Plana formula

AdvancedHelp

(0.002 seconds)

21—30 of 243 matching pages

21: 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.

  • 22: 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. …
    23: 14.28 Sums
    §14.28(ii) Heine’s Formula
    24: Need Help?
  • Finding Things
    • How do I search within DLMF? See Guide to Searching the DLMF.

    • See also the Index or Notations sections.

    • Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the [Uncaptioned image] icon next to each formula, table, figure, and section heading.

  • 25: 5.21 Methods of Computation
    For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …
    26: 12.16 Mathematical Applications
    PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …
    27: 30.10 Series and Integrals
    For product formulas and convolutions see Connett et al. (1993). …
    28: 31.18 Methods of Computation
    Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see Laĭ (1994) and Lay et al. (1998). …
    29: 25.10 Zeros
    §25.10(ii) Riemann–Siegel Formula
    Sign changes of Z ( t ) are determined by multiplying (25.9.3) by exp ( i ϑ ( t ) ) to obtain the Riemann–Siegel formula:
    25.10.3 Z ( t ) = 2 n = 1 m cos ( ϑ ( t ) t ln n ) n 1 / 2 + R ( t ) , m = t / ( 2 π ) ,
    Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of ζ ( s ) in the critical strip are on the critical line (van de Lune et al. (1986)). …
    30: 24.17 Mathematical Applications
    Euler–Maclaurin Summation Formula
    Boole Summation Formula