infinite sequences

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1: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►This sequence converges pointwise to a function $f(z)$ if …The sequence converges uniformly on $S$, if for every $ϵ>0$ there exists an integer $N$, independent of $z$, such that … ►

§1.9(vii) Inversion of Limits

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1.9.66 $$z_{p,q}=\sum _{m=0}^p\sum _{n=0}^q{\zeta }_{m,n}.$$

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Symbols:

$z$: variable, $m$: nonnegative integer, $n$: nonnegative integer and ${\zeta }_{p,q}$: sum

Permalink:

http://dlmf.nist.gov/1.9.E66

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(vii), §1.9(vii), §1.9 and Ch.1

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2: 17.12 Bailey Pairs

… ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …

3: 28.29 Definitions and Basic Properties

… ►To every equation (28.29.1), there belong two increasing infinite sequences of real eigenvalues: …

4: 1.10 Functions of a Complex Variable

… ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. …

5: 2.1 Definitions and Elementary Properties

… ►In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. … ►Let ${\varphi }_s(x)$, $s=0,1,2,\mathrm{\dots }$, be a sequence of functions defined in $𝐗$ such that for each $s$ …where $c$ is a finite, or infinite, limit point of $𝐗$. Then $\{{\varphi }_s(x)\}$ is an asymptotic sequence or scale. Suppose also that $f(x)$ and $f_s(x)$ satisfy …

6: 2.10 Sums and Sequences

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The first infinite integral in (2.10.2) converges.

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7: 1.16 Distributions

… ►A test function is an infinitely differentiable function of compact support. … ►More generally, if $\alpha (x)$ is an infinitely differentiable function, then …We say that a sequence of distributions $\{{\mathrm{\Lambda }}_n\}$ converges to a distribution $\mathrm{\Lambda }$ in ${𝒟}^{\ast }$ if … ►Suppose $f(x)$ is infinitely differentiable except at $x_0$, where left and right derivatives of all orders exist, and … ►A sequence of tempered distributions ${\mathrm{\Lambda }}_n$ converges to $\mathrm{\Lambda }$ in ${𝒯}^{\ast }$ if …

8: Bibliography O

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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OEIS Foundation, Inc., Highland Park, New Jersey.

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Notes:

A tool for finding and identifying integer sequences.

External Links:

On-Line Encyclopedia of Integer Sequences

Referenced by:

5th item.

Encodings:

BibTeX

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A closed form solution of the $s$-wave Bethe-Goldstone equation with an infinite repulsive core. J. Math. Phys. 27 (4), pp. 1154–1158.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

BibTeX

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9: Bibliography C

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Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

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Notes:

Double-precision Fortran, maximum accuracy 24S.

External Links:

ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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Department of Computer Science, University of Victoria, Canada.

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Notes:

Permutations, set and integer partitions, sequences, graphs, and trees.

External Links:

Combinatorial Object Server

Referenced by:

1st item.

Encodings:

BibTeX

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Advanced Combinatorics: The Art of Finite and Infinite Expansions. enlarged edition, D. Reidel Publishing Co., Dordrecht.

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Notes:

Revised and enlarged English translation by D. Reidel of Analyse Combinatoire, Tomes I, II, Presses Universitaires de France, Paris, 1970.

External Links:

ISBN 90-277-0441-4, MathReview, ZentralBlatt

Referenced by:

§26.3(i), §26.3(iii), §26.3(iv), §26.3(v), §26.4(i), §26.4(ii), §26.4(iii), §26.5(i), §26.5(ii), §26.5(iii), §26.6(i), §26.6(ii), §26.6(iii), §26.7(i), §26.7(ii), §26.7(iii), §26.8(i), §26.8(ii), §26.8(iv), §26.8(v), Ch.26.

Encodings:

BibTeX

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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

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10: 6.13 Zeros

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6.13.1 $$x_0=\mathrm{0.37250\hspace{0.33em}74107\hspace{0.33em}81366\hspace{0.33em}63446\hspace{0.33em}19918\hspace{0.33em}66580}\mathrm{\dots }.$$

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Symbols:

$x$: real variable

Notes:

For more digits see OEIS Sequence A091723; see also Sloane (2003).

Referenced by:

§6.13

Permalink:

http://dlmf.nist.gov/6.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.13 and Ch.6

► $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ each have an infinite number of positive real zeros, which are denoted by $c_k$, $s_k$, respectively, arranged in ascending order of absolute value for $k=0,1,2,\mathrm{\dots }$. …