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An algorithm for the solution of certain differential-difference equations of advanced type


Author: G. Greaves
Journal: Math. Comp. 38 (1982), 237-247
MSC: Primary 65R20; Secondary 10H30
DOI: https://doi.org/10.1090/S0025-5718-1982-0637302-7
MathSciNet review: 637302
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Abstract: The differential-difference problem

$\displaystyle \frac{d}{{ds}}\;\{ {sJ(s)} \} + J(s + 1) = 0\;(\operatorname{Re} (s) > 0);\quad sJ(s) \to 1\;{\text{as}}\;s \to 0,$

which arises in the theory of numbers, is studied and an algorithm developed for the simultaneous numerical approximation of the derivatives $ {J^{(n)}}(s)$. An error bound is established for the process described and its numerical performance and stability are discussed.

References [Enhancements On Off] (What's this?)

  • [1] R. Bellman & K. L. Cooke, Differential-Difference Equations, Academic Press, London, 1963. MR 0147745 (26:5259)
  • [2] N. G. de Bruijn, ``On the number of uncancelled elements in the sieve of Eratosthenes,'' Nederl. Akad. Wetensch. Proc., v. 53, 1950, pp. 803-812 = Indag. Math., v. 12, 1950, pp. 247-256. MR 0035785 (12:11d)
  • [3] N. G. de Bruijn, ``On the number of positive integers $ \leqslant x$ and free of prime factors $ > y$,'' Nederl. Akad. Wetensch. Proc. Ser. A, v. 54, 1951, pp. 50-60. MR 0046375 (13:724e)
  • [4] K. Dickman, ``On the frequency of numbers containing prime factors of a certain relative magnitude,'' Ark. Mat., v. 22A, 1930, pp. 1-14.
  • [5] G. Greaves, ``A weighted sieve of Brun's type,'' Acta Arith., v. 40, 1981, pp. 301-335. MR 664619 (83m:10082)
  • [6] H. Iwaniec, ``Rosser's Sieve,'' Acta Arith., v. 36, 1980, pp. 171-202. MR 581917 (81m:10086)
  • [7] H. Iwaniec, J. van der Lune & H. J. J. te Riele, ``The limits of Buchstab's iteration sieve,'' Nederl. Akad. Wetensch. Proc. Ser. A, v. 83, 1980, pp. 409-417.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0637302-7
Article copyright: © Copyright 1982 American Mathematical Society

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