# Taylor series

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## 1—10 of 17 matching pages

##### 1: 5.7 Series Expansions
###### §5.7(i) Maclaurin and TaylorSeries
For 15D numerical values of $c_{k}$ see Abramowitz and Stegun (1964, p. 256), and for 31D values see Wrench (1968).
##### 2: 3.7 Ordinary Differential Equations
###### §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 $\mathbb{C}$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and $\mathrm{Ai}'\left(z\right)$ 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 $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, $\mathrm{Bi}'\left(z\right)$.

• ##### 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) $m$ ($\geq\!1$) 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 $f(z)=\sum_{n=-\infty}^{\infty}f_{n}z^{n},$ $0<|z|.
What is the asymptotic behavior of $f_{n}$ as $n\to\infty$ or $n\to-\infty$? 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, …
##### 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 $f(\alpha,w)$ in a Taylor series centered at the peak value $w=a$ of the exponential factor in the integrand: …