# Taylor series

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## 1—10 of 18 matching pages

##### 1: 5.7 Series Expansions

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###### §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
$$\mathrm{ln}\mathrm{\Gamma}\left(1+z\right)=-\mathrm{ln}\left(1+z\right)+z(1-\gamma )+\sum _{k=2}^{\mathrm{\infty}}{(-1)}^{k}(\zeta \left(k\right)-1)\frac{{z}^{k}}{k},$$
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##### 2: 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. …##### 3: 9.19 Approximations

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Corless et al. (1992) describe a method of approximation based on subdividing $\u2102$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime}\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime}\left(z\right)$ 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 $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime}\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime}\left(z\right)$.

##### 4: 1.10 Functions of a Complex Variable

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►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*) $m$ ($\ge 1$) 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 $$, and in this case the Laurent series becomes the Taylor series. …##### 5: 1.5 Calculus of Two or More Variables

##### 6: 2.10 Sums and Sequences

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2.10.25
$$f(z)=\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}{f}_{n}{z}^{n},$$
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►What is the asymptotic behavior of ${f}_{n}$ as $n\to \mathrm{\infty}$ or $n\to -\mathrm{\infty}$? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion?
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##### 7: 19.19 Taylor and Related Series

###### §19.19 Taylor and Related Series

…##### 8: Bibliography D

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The Taylor Series.
Oxford University Press, Oxford.
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##### 9: 2.4 Contour Integrals

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►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 $t={t}_{0}$.
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##### 10: 2.3 Integrals of a Real Variable

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►We now expand $f(\alpha ,w)$ in a Taylor series centered at the peak value $w=a$ of the exponential factor in the integrand:
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