Laurent series

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1: 4.33 Maclaurin Series and Laurent Series

§4.33 Maclaurin Series and Laurent Series

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2: 4.19 Maclaurin Series and Laurent Series

§4.19 Maclaurin Series and Laurent Series

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3: 23.17 Elementary Properties

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§23.17(ii) Power and Laurent Series

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4: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

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5: 1.10 Functions of a Complex Variable

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§1.10(iii) Laurent Series

… ►This singularity is removable if $a_n=0$ for all , and in this case the Laurent series becomes the Taylor series. … ►The coefficient $a_{-1}$ of ${(z-z_0)}^{-1}$ in the Laurent series for $f(z)$ is called the residue of $f(z)$ at $z_0$, and denoted by ${res}_{z=z_0}[f(z)]$, $\underset{z=z_0}{res}[f(z)]$, or (when there is no ambiguity) $res[f(z)]$. …

6: 25.2 Definition and Expansions

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25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurentseries

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

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

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7: 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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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

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8: 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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Orthogonal Laurent polynomials. Nederl. Akad. Wetensch. Indag. Math. 48 (1), pp. 17–36.

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External Links:

ISSN 0019-3577, Document, MathReview (W. J. Thron), ZentralBlatt

Referenced by:

§18.33(v).

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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9: Bibliography V

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

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Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.

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External Links:

ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

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An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.

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External Links:

ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.60(iii), §10.60(iii).

Encodings:

BibTeX

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Fourier series representation of Ferrers function $𝖯$ .

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External Links:

FullText

Referenced by:

(14.13.1).

Encodings:

BibTeX

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10: Bibliography C

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Some series and bounds for incomplete elliptic integrals. J. Math. and Phys. 40, pp. 125–134.

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External Links:

MathReview (L. Robin), ZentralBlatt

Referenced by:

§19.15, (19.5.1), (19.5.2), (19.5.4), (19.5.4_1), (19.5.4_2), (19.5.4_3), §19.9(ii).

Encodings:

BibTeX

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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:

ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

BibTeX

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Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

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Notes:

Reprinted by Dover, New York, 1950.

External Links:

ZentralBlatt

Referenced by:

§6.16(i).

Encodings:

BibTeX

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On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.

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Notes:

Includes Fortran code

External Links:

Document, ZentralBlatt

Referenced by:

§8.28(vi).

Encodings:

BibTeX

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Basic hypergeometric functions and orthogonal Laurent polynomials. Proc. Amer. Math. Soc. 140 (6), pp. 2075–2089.

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External Links:

ISSN 0002-9939, Link, MathReview (Roelof Koekoek), ZentralBlatt

Referenced by:

§18.33(iv).

Encodings:

BibTeX

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