infinite series expansions

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21: 23.17 Elementary Properties

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§23.17(ii) Power and Laurent Series

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§23.17(iii) Infinite Products

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22: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

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§9.7(ii) Poincaré-Type Expansions

… ►In (9.7.9)–(9.7.12) the $n$th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign. … ►

§9.7(v) Exponentially-Improved Expansions

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23: 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. … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …

24: 1.10 Functions of a Complex Variable

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§1.10(ix) Infinite Products

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§1.10(x) Infinite Partial Fractions

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Mittag-Leffler’s Expansion

… ►Let $F(x,z)$ have a converging power series expansion of the form …

25: Bibliography B

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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Document, ZentralBlatt

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§16.12.

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Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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Document, ZentralBlatt

Referenced by:

§16.6.

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Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

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External Links:

MathReview

Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.

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Infinitely many Stokes smoothings in the gamma function. Proc. Roy. Soc. London Ser. A 434, pp. 465–472.

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ISSN 0962-8444, FullText, MathReview (E. Riekstiņš), ZentralBlatt

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§5.11(ii).

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An analytic continuation of the hypergeometric series. SIAM J. Math. Anal. 18 (3), pp. 884–889.

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ISSN 0036-1410, Document, MathReview (H. M. Srivastava), ZentralBlatt

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§15.15, §15.19(i).

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26: 25.11 Hurwitz Zeta Function

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

§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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§25.11(xii) $a$-Asymptotic Behavior

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27: Bibliography R

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Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.

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ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii), 7th item, §10.77(ix).

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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Link, ISSN 2227-7390, Document

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§8.15.

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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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ISSN 0001-8708, Document, MathReview, ZentralBlatt

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§20.11(iii).

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The problem of an infinite plate under an inclined loading, with tables of the integrals of $\mathrm{Ai}(\pm \mathrm{x})$ and $\mathrm{Bi}(\pm \mathrm{x})$ . Quart. J. Mech. Appl. Math. 7 (1), pp. 1–7.

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Document, MathReview (A. Erdélyi), ZentralBlatt

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§9.16, 1st item.

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Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:

Infinite Series and Products from the Fifteenth to the Twenty-first Century

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ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

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28: Bibliography

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There are infinitely many Carmichael numbers. Ann. of Math. (2) 139 (3), pp. 703–722.

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ISSN 0003-486X, FullText, MathReview (H. L. Montgomery), ZentralBlatt

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§27.12.

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Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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MathReview (R. N. Jain), ZentralBlatt

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§17.11.

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

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§17.12.

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$q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, Vol. 66, Amer. Math. Soc., Providence, RI.

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External Links:

ISBN 0-8218-0716-1, MathReview (David M. Bressoud), ZentralBlatt

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§17.14, §17.14, §17.16, Ch.17, Profile George E. Andrews.

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Table of Definite and Infinite Integrals. Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.

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ISBN 0-444-42151-3, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§1.4(iv), §10.22(vi), §10.43(vi), §11.7(v), §11.9(iv), §12.5(iv), §13.10(vi), §13.23(v), §15.14, §16.20, §16.5, §18.17(ix), §4.10(iii), §4.26(v), §4.40(v), §5.13, §6.14(iii), §7.14(iii), §8.14, §8.19(x).

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29: Bibliography D

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Chebyshev series for the spherical Bessel function $j_l(r)$ . Comput. Phys. Comm. 18 (1), pp. 73–86.

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Document

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§10.76(iii).

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The Taylor Series. Oxford University Press, Oxford.

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Notes:

Reprinted by Dover Publications Inc., New York, 1957

External Links:

MathReview, ZentralBlatt

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§1.9(iii).

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Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London-New York.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(11.11.11), §11.6(i), §11.6(iii), §2.11(v), Ch.2, §8.1, Notations, Notations.

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Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

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Document, ZentralBlatt

Referenced by:

§10.32(iii).

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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ZentralBlatt

Referenced by:

§15.4(ii).

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30: 5.19 Mathematical Applications

… ►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. … ►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …