Taylor series

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1—10 of 18 matching pages

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\Gamma\left(1+z\right)=-\ln\left(1+z\right)+z(1-\gamma)+\sum_{k=2}^{\infty}% (-1)^{k}(\zeta\left(k\right)-1)\frac{z^{k}}{k},

|z|<2

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.33
Referenced by:: §5.21, §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.7(i), §5.7 and Ch.5

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2: 3.7 Ordinary Differential Equations

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§3.7(ii) Taylor-Series Method: Initial-Value Problems

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§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 $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ stored at the nodes. $\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ 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 $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ at the node. Similarly for $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ .

…

4: 1.10 Functions of a Complex Variable

… ►The right-hand side is the Taylor series for

f(z)

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: 1.5 Calculus of Two or More Variables

… ►and the second order term in (1.5.18) is positive definite (negative definite), that is, …

6: 2.10 Sums and Sequences

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2.10.25

f(z)=\sum_{n=-\infty}^{\infty}f_{n}z^{n},

0<|z|<r

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Symbols:: $f(x)$ : analytic function, $f_{n}$ : coefficients and $r$ : radious of annulus
Permalink:: http://dlmf.nist.gov/2.10.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §2.10(iv), §2.10 and Ch.2

►What is the asymptotic behavior of

f_{n}

n\to\infty

n\to-\infty

? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …

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.

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Notes:: Reprinted by Dover Publications Inc., New York, 1957
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.9(iii).
Encodings:: BibTeX

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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)

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: …