Series expansions for the third incomplete elliptic integral via partial fraction decompositions

Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity

We find two convergent series expansions for Legendre's first incomplete elliptic integral $F(λ,k)$ in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square $0<λ,k<1$. Truncated expansions yield asymptotic approximations for $F(λ,k)$ as $λ$ and/or $k$ tend to unity, including the case when logarithmic singularity $λ=k=1$ is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations and numerical examples to illu…

完全第三类 Painlevé 超越函数的椭圆渐近性

Elliptic asymptotics for the complete third Painlevé transcendents

本文利用线性系统的等单值形变与 WKB 分析，证明完全型第三 Painlevé 方程一般解在无穷远附近沿一般方向具有 Jacobi sn 函数形式的 Boutroux 椭圆渐近表示。

第二类不完全椭圆积分在对数奇点附近的收敛展开与界

Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity

本文为勒让德第二类不完全椭圆积分构造了两个由递归初等函数表示的收敛级数展开，并给出各阶显式双边误差界，从而在对数奇点附近得到渐近正确的双边不等式。

Elliptic functions from hypergeometric integrals

As a contribution to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has shown how analogues of the Jacobian elliptic functions may be derived from incomplete hypergeometric integrals in signatures three and four. We determine precisely the signatures in which the Jacobian analogues or their squares are indeed elliptic.

Hyperelliptic uniformization of algebraic curves of the third order

An algebraic function of the third order plays an important role in the problem of asymptotics of Hermite-Padé approximants for two analytic functions with branch points. This algebraic function appears as the Cauchy transform of the limiting measure of the asymptotic distribution of the poles of the approximants. In many cases this algebraic function can be determined by using the given position of the branch points of the functions which are approximated and by the condition that its Abelian integral has purely imaginary periods. In the present paper we obtain a hyperelliptic uniformization …

Numerical computation of real or complex elliptic integrals

Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included.