It is shown that better approximations of tan(x) are obtained if the singular part is taken out, i.e. if we write tan(x) = S(x) + R(x) where S contains the singular part of tan in the sense that the rest R(x) is finite for x → one or several of the points ±π/2,±3π/2,... For example for S(x):= (π/2 - x)-1 + (π/2 + x)-1 we get tan(x) over the whole interval]-π/2,π/2 [as an easily computed part S(x) plus a rest R(x) which is finite (even C), has a rapidly converging power series (in contrast to tan itself), and with low-degree polynomial and rational approximations for any reasonable accuracy. Such approximations are discussed for 1, 2 and 4 singularities taken out. It is also shown that the usual minmax rational approximations over as short an interval as [0,π/4] “contains” both zeroes and singularities of tan — as many as the degrees of numerator and denominator polynomials allow, and with increasing accuracy as these degrees increase. The approximations are examples of a more general situation where a function f to be approximated is split in two parts f = fp + fa, with fp as a simple “preconditioning” part, which mainly serves to give the “accuracy” part fa certain desirable qualities.


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  1. [1]
    Jolley, L.B.W.: Summation of Series. New York, Dover Publications 1961. The table of Bernoulli numbers has been taken from “Report of the British Association for the Advancement of Science, 1877”.Google Scholar
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    Abramowitz, Milton and Irene A. Stegun (ed.): Handbook of Mathematical Functions. New York, Dover Publications 1965. Most of the tables I was interested in has been taken from H. T. Davis: Tables of the higher mathematical functions, vol II. Bloomington, Ind., Princi-pia Press 1935 — but I have only been able to locate vol. I.Google Scholar
  3. [3]
    Pedersen, Poul Wulff: Rapidly converging series for tan and cot, and some related approximations. To appear.Google Scholar
  4. [4]
    Hart, John F. (ed.): Computer Approximations. New York, Wiley 1968. Poul Wulff Pedersen D.T.H., The Technical University of Denmark Dept. of Mathematics, Bygn. 303Google Scholar

Copyright information

© Springer Basel AG 1980

Authors and Affiliations

  • Poul Wulff Pedersen
    • 1
  1. 1.Dept. of MathematicsD.T.H., The Technical University of DenmarkLyngbyDenmark

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