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based on Sinc functions

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1: 3.3 Interpolation
For interpolation of a bounded function f on the cardinal function of f is defined by …
2: Bibliography S
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • 3: 7 Error Functions, Dawson’s and Fresnel Integrals
    Chapter 7 Error Functions, Dawson’s and Fresnel Integrals
    4: 3.4 Differentiation
    For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). …
    §3.4(ii) Analytic Functions
    3.4.18 1 k ! f ( k ) ( x 0 ) = 1 2 π r k 0 2 π f ( x 0 + r e i θ ) e i k θ d θ .
    3.4.19 1 k ! = 1 2 π r k 0 2 π e r cos θ cos ( r sin θ k θ ) d θ .
    5: 25.11 Hurwitz Zeta Function
    25.11.30 ζ ( s , a ) = Γ ( 1 s ) 2 π i ( 0 + ) e a z z s 1 1 e z d z , s 1 , a > 0 ,
    25.11.37 k = 1 ( 1 ) k k ζ ( n k , a ) = n ln Γ ( a ) + ln ( j = 0 n 1 Γ ( a e ( 2 j + 1 ) π i / n ) ) , n = 2 , 3 , 4 , , a 1 .
    6: 33.11 Asymptotic Expansions for Large ρ
    33.11.1 H ± ( η , ρ ) e ± i θ ( η , ρ ) k = 0 ( a ) k ( b ) k k ! ( ± 2 i ρ ) k ,
    7: 4.45 Methods of Computation
    Logarithms
    and since | y | 1 2 ln 10 = 1.15 , e y can be computed straightforwardly from (4.2.19).
    Trigonometric Functions
    and since | θ | 1 2 π = 1.57 , sin θ and cos θ can be computed straightforwardly from (4.19.1) and (4.19.2). …
    Inverse Trigonometric Functions
    8: 25.15 Dirichlet L -functions
    §25.15 Dirichlet L -functions
    When χ is a primitive character (mod k ) the L -functions satisfy the functional equation: …
    §25.15(ii) Zeros
    Since L ( s , χ ) 0 if s > 1 , (25.15.5) shows that for a primitive character χ the only zeros of L ( s , χ ) for s < 0 (the so-called trivial zeros) are as follows: …
    9: 1.16 Distributions
    Λ : 𝒟 ( I ) is called a distribution, or generalized function, if it is a continuous linear functional on 𝒟 ( I ) , that is, it is a linear functional and for every ϕ n ϕ in 𝒟 ( I ) , … Since δ x 0 is the Lebesgue–Stieltjes measure μ α corresponding to α ( x ) = H ( x x 0 ) (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of α . … Since 2 π ( δ ) = 1 , we have …Since the quantity on the extreme right of (1.16.41) is equal to 2 π δ , ϕ , as distributions, the result in this equation can be stated as …
    10: 2.3 Integrals of a Real Variable
    For the function Γ see §5.2(i). This result is probably the most frequently used method for deriving asymptotic expansions of special functions. Since q ( t ) need not be continuous (as long as the integral converges), the case of a finite integration range is included. … Another extension is to more general factors than the exponential function. …
  • (d)

    If p ( b ) = , then P 0 ( b ) = 0 and each of the integrals

    2.3.22 e i x p ( t ) P s ( t ) p ( t ) d t , s = 0 , 1 , 2 , ,

    converges at t = b uniformly for all sufficiently large x .