论文标题
在对数奇点附近的第二种椭圆积分的不完整椭圆积分的收敛膨胀和边界
Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity
论文作者
论文摘要
我们在递归计算的基本功能方面找到了Legendre的第二个不完整的椭圆积分$ e（λ，k）$的两个系列扩展。两种扩展都在$（λ，k）$平面的单位正方形的每个点收敛。所提出的扩展的部分总和形成了$ e（λ，k）$的近似值，这些顺序是渐近的，当$λ$和/或$ k $时，它们倾向于统一，包括当两者接近对数奇异性$λ= k = k = 1 $时。在每个近似顺序下给出显式的双面误差边界。这些界限产生了$ e（λ，k）$的越来越精确的渐近纠正双面不平等的序列。为了方便读者，我们进一步介绍了低阶近似和数值示例的明确表达式，以说明其准确性。我们的派生基于串联重排，超几何求和算法以及广泛使用广义超几何函数的特性，包括一些最近的不平等现象。
We find two series expansions for Legendre's second incomplete elliptic integral $E(λ, k)$ in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the $(λ, k)$ plane. Partial sums of the proposed expansions form a sequence of approximations to $E(λ,k)$ which are asymptotic when $λ$ and/or $k$ tend to unity, including when both approach the logarithmic singularity $λ=k=1$ from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for $E(λ, k)$. For the reader's convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities.
