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1: 3.9 Acceleration of Convergence
§3.9(i) Sequence Transformations
All sequences (series) in this section are sequences (series) of real or complex numbers. … , a sequence for which …
§3.9(iv) Shanks’ Transformation
Sequences that are accelerated by Levin’s transformation include logarithmically convergent sequences, i. …
2: 3.12 Mathematical Constants
3.12.1 π = 3.14159 26535 89793 23846
3.12.3 e = 2.71828 18284 59045 23536 ,
3.12.4 γ = 0.57721 56649 01532 86060 ,
3: 1.17 Integral and Series Representations of the Dirac Delta
§1.17(i) Delta Sequences
for a suitably chosen sequence of functions δ n ( x ) , n = 1 , 2 , . Such a sequence is called a delta sequence and we write, symbolically,
1.17.4 lim n δ n ( x ) = δ ( x ) , x .
An example of a delta sequence is provided by …
4: 26.22 Software
  • On-Line Encyclopedia of Integer Sequences (website).

  • 5: 3.6 Linear Difference Equations
    Many special functions satisfy second-order recurrence relations, or difference equations, of the form …
    3.6.4 w n / g n 0 , n .
    At the same time we construct a sequence e n , n = 0 , 1 , , defined by
    3.6.8 a n e n = c n e n - 1 - d n p n ,
    3.6.10 p n + 1 w n = p n w n + 1 + e n ,
    6: 2.10 Sums and Sequences
    §2.10 Sums and Sequences
    §2.10(i) Euler–Maclaurin Formula
    §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method
    See also Flajolet and Odlyzko (1990).
    7: 2.1 Definitions and Elementary Properties
    Let ϕ s ( x ) , s = 0 , 1 , 2 , , be a sequence of functions defined in X such that for each s
    2.1.19 ϕ s + 1 ( x ) = o ( ϕ s ( x ) ) , x c in X ,
    Then { ϕ s ( x ) } is an asymptotic sequence or scale. Suppose also that f ( x ) and f s ( x ) satisfy
    2.1.20 f ( x ) = s = 0 n - 1 f s ( x ) + O ( ϕ n ( x ) ) , x c in X ,
    8: 1.9 Calculus of a Complex Variable
    §1.9(v) Infinite Sequences and Series
    This sequence converges pointwise to a function f ( z ) if …
    §1.9(vii) Inversion of Limits
    Double Sequences and Series
    9: 3.8 Nonlinear Equations
    Let z 1 , z 2 , be a sequence of approximations to a root, or fixed point, ζ . …for all n sufficiently large, where A and p are independent of n , then the sequence is said to have convergence of the p th order. … For real functions f ( x ) the sequence of approximations to a real zero ξ will always converge (and converge quadratically) if either: … Starting this iteration in the neighborhood of one of the four zeros ± 1 , ± i , sequences { z n } are generated that converge to these zeros. For an arbitrary starting point z 0 , convergence cannot be predicted, and the boundary of the set of points z 0 that generate a sequence converging to a particular zero has a very complicated structure. …
    10: Guide to Searching the DLMF
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