منابع مشابه
Power series for inverse Jacobian elliptic functions
The 12 inverse Jacobian elliptic functions are expanded in power series by using properties of the symmetric elliptic integral of the first kind. Suitable notation allows three series to include all 12 cases, three of which have been given previously. All coefficients are polynomials in the modulus k that are homogeneous variants of Legendre polynomials. The four series in each of three subsets...
متن کاملFast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions
As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K(m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9 to 19 obtained by truncating Taylor series exp...
متن کاملInequalities for Jacobian elliptic functions and Gauss lemniscate functions
A new proof of inequalities involving Jacobian elliptic functions and their inverse functions are obtained. Similar results for the Gauss lemniscate functions are also established. Upper bounds for the inverse Jacobian elliptic functions and for the Gauss arc lemniscate functions are derived. 2012 Elsevier Inc. All rights reserved.
متن کاملGeneralized Jacobian and Discrete Logarithm Problem on Elliptic Curves
Let E be an elliptic curve over the finite field F_{q}, P a point in E(F_{q}) of order n, and Q a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP. In this paper we reduce the discrete logarithm problem on E[n] to the discrete logarithm on the group F*_{q} , the multiplicative group of nonzero elements of Fq, in the case where n | q...
متن کاملTable of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric R-functions
Any product of real powers of Jacobian elliptic functions can be written in the form csm1 (u, k) ds2 (u, k) nsm3 (u, k). If all three m’s are even integers, the indefinite integral of this product with respect to u is a constant times a multivariate hypergeometric function R−a(b1, b2, b3; x, y, z) with halfodd-integral b’s and −a + b1 + b2 + b3 = 1, showing it to be an incomplete elliptic integ...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1946
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1946-08624-3