Carlson B C
Ames Laboratory and Department of Mathematics, Iowa State University, Ames, IA 50011-3020.
J Res Natl Inst Stand Technol. 2002 Oct 1;107(5):413-8. doi: 10.6028/jres.107.034. Print 2002 Sep-Oct.
Three improvements in reduction and computation of elliptic integrals are made. 1. Reduction formulas, used to express many elliptic integrals in terms of a few standard integrals, are simplified by modifying the definition of intermediate "basic integrals." 2. A faster than quadratically convergent series is given for numerical computation of the complete symmetric elliptic integral of the third kind. 3. A series expansion of an elliptic or hyperelliptic integral in elementary symmetric functions is given, illustrated with numerical coefficients for terms through degree seven for the symmetric elliptic integral of the first kind. Its usefulness for elliptic integrals, in particular, is important.
在椭圆积分的化简和计算方面有三项改进。1. 通过修改中间“基本积分”的定义，简化了用于将许多椭圆积分用少数标准积分表示的化简公式。2. 给出了一个用于第三类完全对称椭圆积分数值计算的比二次收敛更快的级数。3. 给出了椭圆或超椭圆积分在初等对称函数中的级数展开式，并以第一类对称椭圆积分七次项以内的数值系数为例进行说明。尤其重要的是它对椭圆积分的实用性。
