Taylor-series methods
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1: 3.7 Ordinary Differential Equations
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§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. …2: 2.10 Sums and Sequences
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►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.
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►By application of Laplace’s method (§2.3(iii)) and use again of (5.11.7), we obtain
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§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method
… ►What is the asymptotic behavior of as or ? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? … ►See also Flajolet and Odlyzko (1990).3: 9.19 Approximations
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Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of , stored at the nodes. and 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 , at the node. Similarly for , .
4: 2.3 Integrals of a Real Variable
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►This result is probably the most frequently used method for deriving asymptotic expansions of special functions.
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§2.3(iii) Laplace’s Method
… ►§2.3(iv) Method of Stationary Phase
… ►§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method
… ►We now expand in a Taylor series centered at the peak value of the exponential factor in the integrand: …5: 2.4 Contour Integrals
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§2.4(iii) Laplace’s Method
… ►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 and at . … ►For this reason the name method of steepest descents is often used. … ►§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
… ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …6: Bibliography D
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Methods of Numerical Integration.
2nd edition, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL.
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Asymptotic Methods in Analysis.
2nd edition, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam.
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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes.
Phys. Rev. E (3) 57 (1), pp. 252–275.
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Computing spectra of linear operators using the Floquet-Fourier-Hill method.
J. Comput. Phys. 219 (1), pp. 296–321.
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The Taylor Series.
Oxford University Press, Oxford.
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