The elliptic integrals are of interest in various disciplines. While series solutions do exist for complete elliptic integrals, there are no deduced series solutions for Incomplete elliptic integrals, in terms of the special functions. This paper provides novel solutions of the Legendre forms of incomplete elliptic integrals of the first and second kinds in terms of the Euler’s gamma functions. The paper also proposes new solutions to inverse trigonometric functions, which has never been known till date.

中文翻译：

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Legendre第一类和第二类不完全椭圆积分和p椭圆积分的新解

椭圆积分在各个学科中都很受关注。虽然完全椭圆积分确实存在级数解，但就特殊函数而言，不完全椭圆积分没有推导出的级数解。本文根据欧拉伽马函数提供了第一类和第二类不完全椭圆积分的勒让德形式的新解。该论文还提出了反三角函数的新解决方案，这是迄今为止未知的。