If you need an accessible version of this item please contact JSTOR User Support

On the Series Expansion Method for Computing Incomplete Elliptic Integrals of the First and Second Kinds

H. Van de Vel
Mathematics of Computation
Vol. 23, No. 105 (Jan., 1969), pp. 61-69
DOI: 10.2307/2005054
Stable URL: http://www.jstor.org/stable/2005054
Page Count: 9
  • Download PDF
  • Cite this Item

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support
On the Series Expansion Method for Computing Incomplete Elliptic Integrals of the First and Second Kinds
Preview not available

Abstract

In the present paper an attempt is made to improve the series expansion method for computing the incomplete integrals F(φ, k) and E(φ, k). Therefore the following three pairs of series covering the region $-1 \leqq k \leqq 1, 0 \leqq \phi < \pi/2$ are used: series obtained by a straightforward binomial expansion of the integrands, series valid for $k'^2 \tan^2 \phi < 1$, and new series which converge for $\phi > \pi/4$ and for all values of k. Terms of the last two pairs of series can be generated by means of the same recurrence relations, so that the coding of the whole is not longer than that for similar methods using only two pairs of series. Any degree of accuracy can be obtained. In general the method is a little bit slower than Bulirsch' calculation procedures which are based on the Landen transformation, but it works more quickly in case of large values of k2 and/or φ. The new series introduced are also represented in trigonometric form, and the double passage to the limit k2 → 1, φ → π/2 is discussed.

Page Thumbnails

  • Thumbnail: Page 
61
    61
  • Thumbnail: Page 
62
    62
  • Thumbnail: Page 
63
    63
  • Thumbnail: Page 
64
    64
  • Thumbnail: Page 
65
    65
  • Thumbnail: Page 
66
    66
  • Thumbnail: Page 
67
    67
  • Thumbnail: Page 
68
    68
  • Thumbnail: Page 
69
    69