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1: 5.7 Series Expansions
§5.7(i) Maclaurin and Taylor Series
For 15D numerical values of c k see Abramowitz and Stegun (1964, p. 256), and for 31D values see Wrench (1968).
5.7.3 ln Γ ( 1 + z ) = - ln ( 1 + z ) + z ( 1 - γ ) + k = 2 ( - 1 ) k ( ζ ( k ) - 1 ) z k k , | z | < 2 .
2: 3.7 Ordinary Differential Equations
§3.7(ii) Taylor-Series Method: Initial-Value Problems
§3.7(iii) Taylor-Series Method: Boundary-Value Problems
It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). … The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …
3: 9.19 Approximations
  • Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of Ai ( z ) , Ai ( z ) stored at the nodes. Ai ( z ) and Ai ( z ) are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of Ai ( z ) , Ai ( z ) at the node. Similarly for Bi ( z ) , Bi ( z ) .

  • 4: 1.10 Functions of a Complex Variable
    The right-hand side is the Taylor series for f ( z ) at z = z 0 , and its radius of convergence is at least R . … An analytic function f ( z ) has a zero of order (or multiplicity) m ( 1 ) at z 0 if the first nonzero coefficient in its Taylor series at z 0 is that of ( z - z 0 ) m . … This singularity is removable if a n = 0 for all n < 0 , and in this case the Laurent series becomes the Taylor series. …
    5: 2.10 Sums and Sequences
    2.10.25 f ( z ) = n = - f n z n , 0 < | z | < r .
    What is the asymptotic behavior of f n as n or n - ? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …
    6: 1.5 Calculus of Two or More Variables
    and the second-order term in (1.5.18) is positive definite (negative definite), that is, …
    7: 19.19 Taylor and Related Series
    §19.19 Taylor and Related Series
    8: Bibliography D
  • P. Dienes (1931) The Taylor Series. Oxford University Press, Oxford.
  • 9: 2.4 Contour Integrals
    and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of p ( t ) and q ( t ) at t = t 0 . …
    10: 2.3 Integrals of a Real Variable
    We now expand f ( α , w ) in a Taylor series centered at the peak value w = a of the exponential factor in the integrand: …