Maclaurin

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21: 13.2 Definitions and Basic Properties

… ►The first two standard solutions are: …

22: 13.29 Methods of Computation

… ►Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …

23: Bibliography S

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Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective. Numer. Math. 98 (2), pp. 371–387.

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External Links:

ISSN 0029-599X, Document, MathReview (Iulian Coroian), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

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Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp. 81 (280), pp. 2159–2173.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (Vittoria Demichelis), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

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Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities. Constr. Approx. 36 (3), pp. 331–352.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Razvan A. Mezei), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

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24: Bibliography F

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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

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Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

Encodings:

BibTeX

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25: Bibliography H

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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External Links:

ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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26: 3.10 Continued Fractions

… ►For example, by converting the Maclaurin expansion of $\arctan z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in ℂ$ except on the branch cuts from $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$ and $-\mathrm{i}$ to $-\mathrm{i}\mathrm{\infty }$. …

27: 13.9 Zeros

… ►The same results apply for the $n$th partial sums of the Maclaurin series (13.2.2) of $M(a,b,z)$. …

28: 31.3 Basic Solutions

… ►If the other exponent is not a positive integer, that is, if $\gamma \ne 0,-1,-2,\mathrm{\dots }$, then from §2.7(i) it follows that $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ exists, is analytic in the disk , and has the Maclaurin expansion …

29: 3.11 Approximation Techniques

… ►However, in general (3.11.11) affords no advantage in $ℂ$ for numerical purposes compared with the Maclaurin expansion of $f(z)$. … ►Thus if $b_0\ne 0$, then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in $z^{p+q}$. …

30: 3.6 Linear Difference Equations

… ►We apply the algorithm to compute ${𝐄}_n\left(1\right)$ to 8S for the range $n=1,2,\mathrm{\dots },10$, beginning with the value ${𝐄}_0\left(1\right)=-\mathrm{0.56865\hspace{0.33em}\hspace{0.17em}663}$ obtained from the Maclaurin series expansion (§11.10(iii)). …