# infinite sequences

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## 1—10 of 18 matching pages

##### 1: 1.9 Calculus of a Complex Variable
###### §1.9(v) InfiniteSequences and Series
This sequence converges pointwise to a function $f(z)$ if …The sequence converges uniformly on $S$, if for every $\epsilon>0$ there exists an integer $N$, independent of $z$, such that …
##### 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 $\phi_{s}(x)$, $s=0,1,2,\dots$, be a sequence of functions defined in $\mathbf{X}$ such that for each $s$ …where $c$ is a finite, or infinite, limit point of $\mathbf{X}$. Then $\{\phi_{s}(x)\}$ is an asymptotic sequence or scale. Suppose also that $f(x)$ and $f_{s}(x)$ satisfy …
##### 6: 2.10 Sums and Sequences
• (c)

The first infinite integral in (2.10.2) converges.

• ##### 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 $\{\Lambda_{n}\}$ converges to a distribution $\Lambda$ in $\mathcal{D}^{*}$ 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 $\Lambda_{n}$ converges to $\Lambda$ in $\mathcal{T}^{*}$ if …
##### 8: Bibliography O
• S. Okui (1974) Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.
• S. Okui (1975) Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.
• On-Line Encyclopedia of Integer Sequences (website) OEIS Foundation, Inc., Highland Park, New Jersey.
• M. K. Ong (1986) A closed form solution of the $s$-wave Bethe-Goldstone equation with an infinite repulsive core. J. Math. Phys. 27 (4), pp. 1154–1158.
• ##### 9: Bibliography C
• W. J. Cody (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.
• Combinatorial Object Server (website) Department of Computer Science, University of Victoria, Canada.
• L. Comtet (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. enlarged edition, D. Reidel Publishing Co., Dordrecht.
• 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.
• ##### 10: 6.13 Zeros
6.13.1 $x_{0}=0.37250\;74107\;81366\;63446\;19918\;66580\dots.$
$\operatorname{Ci}\left(x\right)$ and $\operatorname{si}\left(x\right)$ 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 $k=0,1,2,\dots$. …