Maclaurin series

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21: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

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22: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

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23: 25.2 Definition and Expansions

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§25.2(ii) Other Infinite Series

… ►For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. … ►

§25.2(iii) Representations by the Euler–Maclaurin Formula

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25.2.8 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{x^{s+1}}dx,$$ $\mathrm{\Re }s>0$, $N=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1976, (25), p. 269)

Proof sketch:

Derivable from Apostol (1976, Theorem 12.21, p. 269) by setting $a=1$ and $N\mapsto N-1$.

A&S Ref:

23.2.9

Referenced by:

(25.2.9)

Permalink:

http://dlmf.nist.gov/25.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

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25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Proof sketch:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

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24: 15.19 Methods of Computation

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§15.19(i) Maclaurin Expansions

►The Gauss series (15.2.1) converges for . … ►However, by appropriate choice of the constant $z_0$ in (15.15.1) we can obtain an infinite series that converges on a disk containing $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. …

25: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

►Every convergent, asymptotic, or formal series … ►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 $-\mathrm{i}\mathrm{\infty }$. … ►We say that it corresponds to the formal power series … ►

Forward Series Recurrence Algorithm

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26: Bibliography E

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Ten Physical Applications of Spectral Zeta Functions. Lecture Notes in Physics. New Series m: Monographs, Vol. 35, Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-60230-5, MathReview (Alfred Actor), ZentralBlatt

Referenced by:

§25.17.

Encodings:

BibTeX

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The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).

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External Links:

ISSN 0334-2700, FullText, MathReview (J. Németh), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

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A new series representation for $\zeta (3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.

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External Links:

ISSN 0002-9890, FullText, MathReview (Christian Radoux), ZentralBlatt

Referenced by:

(25.8.10).

Encodings:

BibTeX

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27: 10.74 Methods of Computation

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§10.74(i) Series Expansions

►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Temme (1997) shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions. … ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). …

28: 25.11 Hurwitz Zeta Function

… ►The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion … ►

§25.11(iii) Representations by the Euler–Maclaurin Formula

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§25.11(iv) Series Representations

… ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►

§25.11(x) Further Series Representations

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29: Bibliography H

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Note on Dr. Vacca’s series for $\gamma$ . Quart. J. Math. 43, pp. 215–216.

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External Links:

ZentralBlatt

Referenced by:

(25.2.4), (25.2.5), §25.2(ii).

Encodings:

BibTeX

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Divergent Series. Clarendon Press, Oxford.

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Notes:

Reprinted by the American Mathematical Society in 2000.

External Links:

MathReview (R. P. Agnew), ZentralBlatt

Referenced by:

§1.15(ii), §1.15(viii).

Encodings:

BibTeX

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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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Note on some hypergeometric series of higher order. J. London Math. Soc. 4, pp. 50–54.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.9.

Encodings:

BibTeX

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30: Bibliography S

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On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.

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External Links:

ISSN 0075-4102, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§5.11(ii).

Encodings:

BibTeX

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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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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

BibTeX

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