Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions
Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions
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雅可比椭圆函数泰勒级数展开系数的性质
DOI：
10.1090/s0025-5718-1976-0391477-3
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发表时间：
1976
影响因子：
2
通讯作者：
A. Schett
中科院分区：
文献类型：
--
作者：
A. Schett
给出了雅可比椭圆函数的泰勒级数展开系数和前15项的表的性质。给出了这些系数与随机化分布的关系。关于Jacobian椭圆函数sn（u, k）、cn（u, k）和dn（u, k）的泰勒级数展开系数，我们所知甚少。这些系数不存在递推公式。只有四到五个主要术语的系列在文献中给出（[1],b[2]）。我们在本文中给出了这些系数的性质，展示了它们与随机化分布之间的关系[3，p. 51]，并给出了前15项的表。我们考虑微分方程d Y1(U) C1y2(U) y3(U) = 0, du (1) d Y2 (-) C2y1(U) y3(U) = 0, du d y3(U) C3y1(U) Y2 (U) = 0du(1)对于C1 = 1, C2 = 1, C3 =的解函数是雅可比椭圆函数v1 = sn(U, k), Y2 = cn(U, k), y3 = dn(U, k)（[1],[2]）。函数y1 Y2 Y3的形式泰勒级数等于co(u (u) 0)n…CJ 21 l ?y(u) = E ！？ Y10Y20Y30 n=0 (2) w (U) n！[b] hi 2 (-3 SI "2 s3r n n！=LE n ！hh2h3 1 C22C3jY10Y20y330J
Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions and tables for the first fifteen leading terms are given. Relations of these coefficients with the randomization distributions are shown. Little is known about the Taylor series expansion coefficients of the Jacobian elliptic functions sn(u, k), cn(u, k) and dn(u, k). No recurrence formula exists for these coefficients. Only four to five leading terms of the series are given in literature ([1], [2]). We present in this paper properties of these coefficients, show relations between them and randomization distributions [3, p. 51], and give tables for the first fifteen leading terms. We consider the differential equations d Y1(U) C1y2(u)y3(U) = 0, du (1) d Y2 ( -) C2y1(u)y3(U) = 0, du d Y3(U) C3y1(u)y2(u) = 0du Solution functions of (1) for C1 = 1, C2 = 1, C3 = are the Jacobian elliptic functions v1 = sn(u, k), Y2 = cn(u, k), Y3 = dn(u, k) ([1], [2]). The formal Taylor series of the functions y1, Y2, Y3 read co(u u0)n ...CJ'21 l? ' y(u) = E ! ? ajj2j3C1 2 3 Y10Y20Y30 n=0 (2) w (U o n! [ b hi 2 (-3 SI "2 s3r n n!=LE n !hh2h3 1 C22C3jY10Y20y330J