Complete Hyper-Elliptic Integrals of the First Kind and the Chebyshev Property (Journal of Nonlinear Modeling and Analysis, 2024)
This paper is devoted to studying the following complete hyper-elliptic integral of the first kind $$J(h)=\oint\limits_{Γ_h}\frac{α_0+α_1x+α_2x^2+α_3x^3}{y}dx,$$ where $α_i\in\mathbb{R}$, $Γ_h$ 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 $α_0=0$ and Chebyshev with accuracy one for $α_i=0\ (i=1,2,3)$.