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1: 1.9 Calculus of a Complex Variable
§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
1.9.66 z p , q = m = 0 p n = 0 q ζ m , n .
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 ϕ s ( x ) , s = 0 , 1 , 2 , , 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 { ϕ 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 α ( x ) is an infinitely differentiable function, then …We say that a sequence of distributions { Λ n } converges to a distribution Λ in 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 Λ n converges to Λ in 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 .
    Ci ( x ) and si ( x ) 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 , . …