ABSTRACT In this paper we treat certain elliptic and hyper-elliptic integrals in a unified way. We introduce a new basis of these integrals coming from certain basis of polynomials and show that the transition matrix between this basis and the traditional monomial basis is certain upper triangular band matrix. This allows us to obtain explicit formulas for the considered integrals. Our approach, specified to elliptic case, is more effective than known recursive procedures for elliptic integrals. We also show that basic integrals enjoy symmetry coming from the action of the dihedral group on a real projective line. This action is closely connected with the properties of homographic transformation of a real projective line. This explains similarities occurring in some formulas in popular tables of elliptic integrals. As a consequence one can reduce the number of necessary formulas in a significant way. We believe that our results will simplify programming and computing the hyper-elliptic integrals in various problems of mathematical physics and engineering.

中文翻译：

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某些实超椭圆积分的新方法

摘要 在本文中，我们统一处理某些椭圆和超椭圆积分。我们从多项式的某些基中引入这些积分的新基，并证明该基与传统单项式基之间的转移矩阵是某个上三角带矩阵。这使我们能够获得所考虑积分的明确公式。我们针对椭圆情况的方法比已知的椭圆积分递归程序更有效。我们还表明，基本积分享有来自二面体群在真实射影线上的作用的对称性。这个动作与实射影线的单应变换性质密切相关。这解释了流行的椭圆积分表中某些公式中出现的相似性。因此，可以显着减少必要公式的数量。我们相信我们的结果将简化数学物理和工程各种问题中超椭圆积分的编程和计算。