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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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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Equation (27.14.2)

27.14.2

\mathit{f}\left(x\right)=\prod_{m=1}^{\infty}(1-x^{m})=\left(x;x\right)_{% \infty},

|x|<1

The representation in terms of $\left(x;x\right)_{\infty}$ was added to this equation.

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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:: §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: 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 formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). … ►and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. …