TY - JOUR
T1 - Complete Hyper-Elliptic Integrals of the First Kind and the Chebyshev Property
AU - Yang , Jihua
JO - Journal of Nonlinear Modeling and Analysis
VL - 3
SP - 431
EP - 446
PY - 2021
DA - 2021/04
SN - 2
DO - http://doi.org/10.12150/jnma.431
UR - https://global-sci.org/intro/article_detail/jnma/18820.html
KW - Complete hyper-elliptic integral of the first kind, Chebyshev,
ECT-system.
AB - <p style="text-align: justify;">This paper is devoted to studying the following complete hyper-elliptic integral of the first kind $$J(h)=\oint\limits_{\Gamma_h}\frac{\alpha_0+\alpha_1x+\alpha_2x^2+\alpha_3x^3}{y}dx,$$ where $\alpha_i\in\mathbb{R}$, $\Gamma_h$ is an oval contained in the level set $\{H(x,y)=h, h\in(-\frac{5}{36},0)\}$ and $H(x,y)=\frac{1}{2}y^2-\frac{1}{4}x^4+\frac{1}{9}x^9$. We show that the 3-dimensional real vector spaces of these integrals are Chebyshev for $\alpha_0=0$ and Chebyshev with accuracy one for $\alpha_i=0\ (i=1,2,3)$.</p>