power-series expansions in r

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11—19 of 19 matching pages

11: Bibliography H

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P. I. Hadži (1976a) 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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R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.

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External Links:: Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §2.5(iii), §2.5(ii).
Encodings:: BibTeX

… ►

E. R. Hansen (1975) A Table of Series and Products. Prentice-Hall, Englewood Cliffs, N.J..

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Notes:: Table erratum: Math. Comp. v.47 (1986), no. 176, p. 767.
External Links:: ZentralBlatt
Referenced by:: §10.23(iv), §10.44(iv), §10.60(iv), §10.65(iv), §11.10(x), §11.7(v), §12.13(ii), §13.11, §14.18(iv), §15.15, §18.18(ix), §24.14(iii), §25.11(xi), §25.8, §4.11, §4.27, §4.41, §5.16, §6.15, §7.15, §8.17(vi).
Encodings:: BibTeX

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G. H. Hardy (1949) 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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P. Henrici (1974) Applied and Computational Complex Analysis. Vol. 1: Power Series—Integration—Conformal Mapping—Location of Zeros. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York.

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Notes:: Reprinted in 1988.
External Links:: ISBN 0-471-37244-7, MathReview (M. Marden), ZentralBlatt
Referenced by:: §1.10(vii), §1.9(vi), Ch.3.
Encodings:: BibTeX

…

12: 30.3 Eigenvalues

… ►In equation (30.3.5) we can also use … ►

§30.3(iv) Power-Series Expansion

►

30.3.8

\lambda^{m}_{n}\left(\gamma^{2}\right)=\sum_{k=0}^{\infty}\ell_{2k}\gamma^{2k},

|\gamma^{2}|<r_{n}^{m}

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Symbols:: $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $\ell_{j}$ : coefficients
Referenced by:: §30.16(i), item
Permalink:: http://dlmf.nist.gov/30.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §30.3(iv), §30.3 and Ch.30

►For values of

r_{n}^{m}

see Meixner et al. (1980, p. 109). …

13: 1.9 Calculus of a Complex Variable

… ►

Powers

… ►

§1.9(v) Infinite Sequences and Series

… ►

§1.9(vi) Power Series

… ►

Operations

… ►Lastly, a power series can be differentiated any number of times within its circle of convergence: …

14: Bibliography R

Bibliography R

… ►

RISC Combinatorics Group (website) Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:: Software packages for hypergeometric and $q$ -hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.
External Links:: RISC Combinatorics Group
Referenced by:: 6th item.
Encodings:: BibTeX

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R. Roy (2011) 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
External Links:: ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt
Referenced by:: Profile Ranjan Roy.
Encodings:: BibTeX

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W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:: McGraw-Hill Series in Higher Mathematics
External Links:: MathReview (F. Smithies), ZentralBlatt
Referenced by:: §1.18(x), §2.6(i).
Encodings:: BibTeX

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W. Rudin (1976) Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:: International Series in Pure and Applied Mathematics
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.4(iii).
Encodings:: BibTeX

…

15: Bibliography B

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P. Barrucand and D. Dickinson (1968) On the Associated Legendre Polynomials. In Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pp. 43–50.

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External Links:: MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §18.30.
Encodings:: BibTeX

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R. Bellman (1961) A Brief Introduction to Theta Functions. Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York.

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External Links:: MathReview (W. J. LeVeque), ZentralBlatt
Referenced by:: §20.10(i), §20.10(ii), §20.11(i), §20.13, §20.5(ii), §20.7(viii), §20.8.
Encodings:: BibTeX

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M. V. Berry (1981) Singularities in Waves and Rays. In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.), Vol. 35, pp. 453–543.

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Referenced by:: §36.14(i).
Encodings:: BibTeX

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L. J. Billera, C. Greene, R. Simion, and R. P. Stanley (Eds.) (1996) Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

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External Links:: ISBN 0-8218-0324-7, MathReview, ZentralBlatt
Referenced by:: §26.19.
Encodings:: BibTeX

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W. G. C. Boyd (1990a) Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.

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External Links:: MathReview (John S. Lew), ZentralBlatt
Referenced by:: §2.8(iv).
Encodings:: BibTeX

…

16: 27.14 Unrestricted Partitions

… ►as a generating function for the function

p\left(n\right)

defined in §27.14(i): … ►Multiplying the power series for

\mathit{f}\left(x\right)

with that for

1/\mathit{f}\left(x\right)

and equating coefficients, we obtain the recursion formula … ►Rademacher (1938) derives a convergent series that also provides an asymptotic expansion for

p\left(n\right)

: … ►This is related to the function

\mathit{f}\left(x\right)

in (27.14.2) by … ►The 24th power of

\eta\left(\tau\right)

in (27.14.12) with

e^{2\pi\mathrm{i}\tau}=x

is an infinite product that generates a power series in

x

with integer coefficients called Ramanujan’s tau function

\tau\left(n\right)

: …

17: 30.16 Methods of Computation

… ►For small

|\gamma^{2}|

we can use the power-series expansion (30.3.8). …If

|\gamma^{2}|

is large we can use the asymptotic expansions in §30.9. … ►If

|\gamma^{2}|

is large, then we can use the asymptotic expansions referred to in §30.9 to approximate

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

. … ►The coefficients

a^{m}_{n,r}(\gamma^{2})

are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). ►A fourth method, based on the expansion (30.8.1), is as follows. …

18: Bibliography C

… ►

B. C. Carlson (2008) 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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J. R. Cash and R. V. M. Zahar (1994) A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.

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External Links:: ISBN 0-8176-3753-2, MathReview (R. M. M. Mattheij), ZentralBlatt
Referenced by:: §3.6(vii).
Encodings:: BibTeX

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T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

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External Links:: MathReview (I. I. Hirschman, Jr.), ZentralBlatt
Referenced by:: §12.16.
Encodings:: BibTeX

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H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

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External Links:: ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt
Referenced by:: §14.28(ii), §14.28(ii).
Encodings:: BibTeX

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R. M. Corless, D. J. Jeffrey, and D. E. Knuth (1997) A sequence of series for the Lambert

W

function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

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External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.
Encodings:: BibTeX

…

19: 18.26 Wilson Class: Continued

… ►

18.26.20

{{}_{2}F_{1}}\left({-y,-y+\beta-\gamma\atop\beta+\delta+1};z\right){{}_{2}F_{1% }}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}^{N}\frac{{\left(-N% \right)_{n}}{\left(\gamma+1\right)_{n}}}{{\left(-\delta-N\right)_{n}}n!}R_{n}% \left(y(y+\gamma+\delta+1);-N-1,\beta,\gamma,\delta\right)z^{n}.

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$ : Racah polynomial, $!$ : factorial (as in $n!$ ), $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

… ►

18.26.21

(1-z)^{y}{{}_{2}F_{1}}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}% ^{N}\frac{{\left(\gamma+1\right)_{n}}{\left(-N\right)_{n}}}{{\left(-\delta-N% \right)_{n}}n!}\*R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)z^{n}.

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : dual Hahn polynomial, $!$ : factorial (as in $n!$ ), $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.26(iv)
Permalink:: http://dlmf.nist.gov/18.26.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

… ►For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998). ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.