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Mathematics of Computation

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Plana's summation formula for $ \sum\sp \infty\sb {m=1,3,\cdots}m\sp {-2}\sin(m\alpha),m\sp {-3}\cos(m\alpha),m\sp {-2}A\sp m,m\sp {-3}A\sp m$

Authors: Kevin M. Dempsey, Dajin Liu and John P. Dempsey
Journal: Math. Comp. 55 (1990), 693-703
MSC: Primary 65B10
MathSciNet review: 1035929
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Abstract: The four series

\begin{displaymath}\begin{array}{*{20}{c}} {\sum\limits_0^\infty {\sin (2k + 1)\... ...mits_0^\infty {{A^{2k + 1}}/{{(2k + 1)}^3}} } } \\ \end{array} \end{displaymath}

are very slowly convergent for $ 0 \leq \alpha \leq \pi $ and as $ A \to {1^ - }$. Direct summation involves thousands of terms to get the accuracy desired. Plana's summation formula along with Romberg's method of integration significantly and consistently improves the convergence and accuracy for the above series.

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

  • [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Applied Mathematics Series No. 55, National Bureau of Standards, 1964.
  • [2] L. M. Delves and J. L. Mohamed, Computational methods for integral equations, Cambridge University Press, Cambridge, 1985. MR 837187
  • [3] J. P. Dempsey, L. M. Keer, N. B. Patel, and M. L. Glasser, Contact between plates and unilateral supports, ASME J. Appl. Mech. 51 (1984), 324-328.
  • [4] J. P. Dempsey and Hui Li, Rectangular plates on unilateral edge supports: Part 1--Theory and numerical analysis, ASME J. Appl. Mech. 53 (1986), 146-150.
  • [5] -, Rectangular plates on unilateral edge supports: Part 2--Implementation; concentrated and uniform loading, ASME J. Appl. Mech. 53 (1986), 151-156.
  • [6] M. Lawrence Glasser, The summation of series, SIAM J. Math. Anal. 2 (1971), 595–600. MR 0303156
  • [7] -, personal communication, Clarkson University, 1988.
  • [8] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, Academic Press, New York, 1965.
  • [9] N. Grossman, personal communication, University of California at Los Angeles, 1988.
  • [10] E. R. Hansen, A table of series and products, Prentice-Hall, Englewood Cliffs, N.J., 1975.
  • [11] Peter Henrici, Elements of numerical analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0166900
  • [12] Peter Henrici, Applied and computational complex analysis, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros; Pure and Applied Mathematics. MR 0372162
  • [13] Peter Henrici, Essentials of numerical analysis with pocket calculator demonstrations, John Wiley & Sons, Inc., New York, 1982. MR 655251
  • [14] David Levin, Development of non-linear transformations of improving convergence of sequences, Internat. J. Comput. Math. 3 (1973), 371–388. MR 0359261
  • [15] L. Lewin, Dilogarithms and associated functions, Foreword by J. C. P. Miller, Macdonald, London, 1958. MR 0105524
  • [16] E. Lindelöf, Le calcul des résidus et ses applications à la théorie des fonctions, Gauthier-Villars, 1905; reprint, Chelsea, New York, 1947.

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Article copyright: © Copyright 1990 American Mathematical Society