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Mathematics of Computation

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A rapidly convergent series for computing $ \psi (z)$ and its derivatives

Author: Peter McCullagh
Journal: Math. Comp. 36 (1981), 247-248
MSC: Primary 65D20; Secondary 33A15
MathSciNet review: 595057
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Abstract: We derive a series expansion for $ \psi (z)$ in which the terms of the expansion are simple rational functions of z. From a computational viewpoint, the new series is of interest in that it converges for all z not necessarily real valued, and is particularly rapid for values of z near the origin. From a mathematical viewpoint the series is of interest in that, although $ \psi (z)$ has poles at the negative integers and zero, the series is uniformly convergent in any finite interval $ a < \operatorname{Re} (z) < b$.

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

  • [1] M. Abramowitz & I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970.
  • [2] Y. L. Luke, The Special Functions and Their Approximations, Academic Press, New York, 1969.
  • [3] Yudell L. Luke, Mathematical functions and their approximations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0501762

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Article copyright: © Copyright 1981 American Mathematical Society