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Maclaurin series

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11: Bibliography F
  • W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.
  • 12: 13.2 Definitions and Basic Properties
    The first two standard solutions are: …
    13: 13.9 Zeros
    The same results apply for the n th partial sums of the Maclaurin series (13.2.2) of M ( a , b , z ) . …
    14: 3.6 Linear Difference Equations
    We apply the algorithm to compute 𝐄 n ( 1 ) to 8S for the range n = 1 , 2 , , 10 , beginning with the value 𝐄 0 ( 1 ) = 0.56865  663 obtained from the Maclaurin series expansion (§11.10(iii)). …
    15: 8.12 Uniform Asymptotic Expansions for Large Parameter
    The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at η = 0 , and the Maclaurin series expansion of c k ( η ) is given by …
    16: 18.17 Integrals
    17: 4.13 Lambert W -Function
    4.13.10 W k ( z ) ξ k ln ξ k + n = 1 ( 1 ) n ξ k n m = 1 n [ n n m + 1 ] ( ln ξ k ) m m ! ,
    For large enough | z | the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. In the case of k = 0 and real z the series converges for z e . …
    4.13.11 W ± 1 ( x 0 i ) η ln η + n = 1 1 η n m = 1 n [ n n m + 1 ] ( ln η ) m m ! ,
    18: 2.10 Sums and Sequences
    The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. …
    19: Errata
  • Expansion

    §4.13 has been enlarged. The Lambert W -function is multi-valued and we use the notation W k ( x ) , k , for the branches. The original two solutions are identified via Wp ( x ) = W 0 ( x ) and Wm ( x ) = W ± 1 ( x 0 i ) .

    Other changes are the introduction of the Wright ω -function and tree T -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for d n W d z n , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at z = e 1 in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert W -functions in the end of the section.

  • 20: 3.11 Approximation Techniques
    §3.11(ii) Chebyshev-Series Expansions
    Summation of Chebyshev Series: Clenshaw’s Algorithm
    However, in general (3.11.11) affords no advantage in for numerical purposes compared with the Maclaurin expansion of f ( z ) . … be a formal power series. …Thus if b 0 0 , then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in z p + q . …