representations by Euler–Maclaurin formula

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1: 25.2 Definition and Expansions

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

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2: 25.11 Hurwitz Zeta Function

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

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3: Errata

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Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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Expansion

§4.13 has been enlarged. The Lambert $W$ -function is multi-valued and we use the notation $W_{k}\left(x\right)$ , $k\in\mathbb{Z}$ , for the branches. The original two solutions are identified via $\operatorname{Wp}\left(x\right)=W_{0}\left(x\right)$ and $\operatorname{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$ .

Other changes are the introduction of the Wright $\omega$ -function and tree $T$ -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{{\mathrm{d}}^{n}W}{{\mathrm{d}z}^{n}}$ , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$ -functions in the end of the section.

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Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

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Equations (15.6.1)–(15.6.9)

The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ has been added. In (15.6.6), the constraint $|\operatorname{ph}\left(-z\right)|<\pi$ has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when $|\operatorname{ph}\left(1-z\right)|<\pi$ , except (15.6.6) which holds for $|\operatorname{ph}\left(-z\right)|<\pi$ .”, has been removed.

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Equation (5.11.8)

It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for $\operatorname{Ln}\Gamma\left(z+h\right)$ is valid for $h$ $(\in\mathbb{C})$ ; originally it was unnecessarily restricted to $[0,1]$ .

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4: Bibliography E

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G. P. Egorychev (1984) Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFadden, Translation edited by Lev J. Leifman
External Links:: ISBN 0-8218-4512-8, MathReview, ZentralBlatt
Referenced by:: §15.17(iv).
Encodings:: BibTeX

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D. Elliott (1998) 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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T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

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External Links:: Document, MathReview (S. Chowla), ZentralBlatt
Referenced by:: §20.12(i).
Encodings:: BibTeX

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L. Euler (1768) Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.

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Referenced by:: §25.12(i).
Encodings:: BibTeX

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J. A. Ewell (1990) 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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5: Bibliography B

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Á. Baricz and T. K. Pogány (2013) Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:: ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt
Referenced by:: §11.7(v), §11.7(v).
Encodings:: BibTeX

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W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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External Links:: ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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External Links:: Document, MathReview (T. M. Apostol), ZentralBlatt
Referenced by:: §24.16(ii).
Encodings:: BibTeX

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B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.16(iii), §24.8(ii).
Encodings:: BibTeX

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P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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External Links:: ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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6: 9.12 Scorer Functions

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§9.12(v) Connection Formulas

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§9.12(vi) Maclaurin Series

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§9.12(vii) Integral Representations

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7: Bibliography F

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P. Flajolet and B. Salvy (1998) Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.

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External Links:: ISSN 1058-6458, Link, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: (25.16.13), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

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J. P. M. Flude (1998) The Edmonds asymptotic formulas for the

3j

and

6j

symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

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External Links:: ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt
Referenced by:: §34.8.
Encodings:: BibTeX

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W. B. Ford (1960) 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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G. Freud (1976) On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:: MathReview (A. G. Law), ZentralBlatt
Referenced by:: §18.32, §32.15.
Encodings:: BibTeX

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8: 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. … ►For

U\left(a,b,z\right)

and

W_{\kappa,\mu}\left(z\right)

we may integrate along outward rays from the origin in the sectors

\tfrac{1}{2}\pi<|\operatorname{ph}z|<\tfrac{3}{2}\pi

, with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). … ►

§13.29(iii) Integral Representations

►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. … ►

13.29.8

w(n)\sim\frac{\sqrt{\pi}e^{\frac{1}{2}z}z^{\frac{1}{4}(4a-2b+1)}}{\Gamma\left(% a\right)\Gamma\left(a+1-b\right)}n^{\frac{1}{4}(4a-2b-3)}e^{-2\sqrt{nz}},

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.29.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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9: Bibliography H

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B. Hall (2015) Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:: An elementary introduction
External Links:: ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry
Referenced by:: §1.2(vi).
Encodings:: BibTeX

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M. Hauss (1997) 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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M. Hauss (1998) A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:: ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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External Links:: ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

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F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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External Links:: ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(i).
Encodings:: BibTeX

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10: 18.17 Integrals

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§18.17(ii) Integral Representations for Products

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Ultraspherical

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Legendre

… ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. … ►Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively. …