We find two series expansions for Legendre's second incomplete elliptic integral $E(\lambda, k)$ in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the $(\lambda, k)$ plane. Partial sums of the proposed expansions form a sequence of approximations to $E(\lambda,k)$ which are asymptotic when $\lambda$ and/or $k$ tend to unity, including when both approach the logarithmic singularity $\lambda=k=1$ 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 $E(\lambda, k)$. 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.

翻译：本文找出勒让德第二类不完全椭圆积分$E(\lambda, k)$的两个级数展开式，均以递归计算初等函数表示。这两个展开式在$(\lambda, k)$平面单位正方形内的每一点均收敛。所提展开式的部分和构成$E(\lambda,k)$的近似序列，当$\lambda$和/或$k$趋于1（包括两者从任意方向逼近对数奇点$\lambda=k=1$）时皆具有渐近性。每个近似阶次均给出显式双侧误差界，这些界值产生一列日益精确的$E(\lambda, k)$渐近正确双侧不等式。为方便读者，我们进一步给出低阶近似的显式表达式及数值算例以展示其精度。本文推导基于级数重排、超几何求和算法及广义超几何函数性质（包括近期若干不等式）的广泛运用。