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
…2: 25.11 Hurwitz Zeta Function
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§25.11(iii) Representations by the Euler–Maclaurin Formula
…3: Bibliography E
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Integral Representation and the Computation of Combinatorial Sums.
Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.
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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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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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Institutiones Calculi Integralis.
Opera Omnia (1), Vol. 11, pp. 110–113.
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A new series representation for
.
Amer. Math. Monthly 97 (3), pp. 219–220.
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4: Bibliography B
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Integral representations and summations of the modified Struve function.
Acta Math. Hungar. 141 (3), pp. 254–281.
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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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Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications.
J. Number Theory 7 (4), pp. 413–445.
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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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Riemann zeta function: Rapidly converging series and integral representations.
Appl. Math. Lett. 5 (2), pp. 83–88.
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5: 9.12 Scorer Functions
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►where
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§9.12(v) Connection Formulas
… ►§9.12(vi) Maclaurin Series
… ►§9.12(vii) Integral Representations
… ► …6: Bibliography F
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Euler sums and contour integral representations.
Experiment. Math. 7 (1), pp. 15–35.
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The Edmonds asymptotic formulas for the and symbols.
J. Math. Phys. 39 (7), pp. 3906–3915.
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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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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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7: 13.29 Methods of Computation
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►Although the Maclaurin series expansion (13.2.2) converges for all finite values of , it is cumbersome to use when is large owing to slowness of convergence and cancellation.
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►For and we may integrate along outward rays from the origin in the sectors , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii).
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§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
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8: 18.17 Integrals
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