Further approximations for elliptic integrals

Abstract

Abstract. The present paper develops approximations for the three kinds of elliptic integrals based on the Padé approximations for the square root. The work includes and extends our previous work on the subject to provide efficient approximations over a larger part of the complex k and <¡> planes.1. Introduction. In previous work [1], [2], we gave some approximations for the three kinds of complete and incomplete elliptic integrals which were derived by using Padé approximations for the square root. The expansions are valid in a large sector of the complex plane and are far more powerful than the analogous representations based on the binomial expansion for the square root. Nonetheless, the convergence properties of our previous developments weaken for k2 sin2 </> near unity. In this paper, we develop new representations like those of [1], [2] to correct this deficiency. The present study is suggested by analogous work for power series developments of the first two kinds of elliptic integrals. In this connection, see the recent study by Van de Vel [3] and the references quoted there. For convenience and completeness, we also give the representations previously found. Our proofs are very sketchy as the details are akin to those previously used.

Capacitive Interdigital Sensors for Flexible Enclosures and Wearables

“…plane parallel electrodes [21][22][23][24][25][26][27][28][29] We assume that the sensor is an interdigital one, with the geometry as in Fig. 4(a).…”

“…6. Various approximations are given in the literature for the elliptic integrals [28], [29], [30]. These approximations may be used to further refine the results in this section.…”

Series expansions of symmetric elliptic integrals

“…The series expansions of the classic elliptic integrals have been discussed mainly in order to seek a convenient way to evaluate them by means of the elementary functions such as the logarithm or inverse trigonometric functions [18,21,27,19,32,38,33,2,34]. Nevertheless, these methods are known to be slower than: the standard methods utilizing Landen and/or Gauss transformations [3,4,5,6,26,36].…”