Wronski Determinant of Trigonometric System

Authors : Ufuk KAYA

Pages : 1-8

View : 24 | Download : 10

Publication Date : 2022-03-01

Article Type : Research Paper

Abstract :In this paper, we calculate the Wronskian of the trigonometric system \\[ \\cos{\\lambda_{1}x},\\sin{\\lambda_{1}x},\\cos{\\lambda_{2}x},\\sin{\\lambda_{2}x},\\dots,\\cos{\\lambda_{n}x},\\sin{\\lambda_{n}x} \\] and prove that this system is linearly independent when $\\lambda_{k}\\ne 0$ and $\\lambda_{k}^{2}\\ne \\lambda_{l}^{2}$ for $k\\ne l$, where $\\lambda_{1},\\lambda_{2},\\dots,\\lambda_{n}\\in\\mathbb{C}$, $n\\in\\mathbb{N}$ are constants and $x$ is a complex variable. By using it, we evaluate the determinant below \\[ \\left| \\begin{array}{ccccccc} 1&0&1&0&\\cdots&1&0\\\\ 0&1&0&1&\\cdots&0&1\\\\ \\lambda_{1}&0&\\lambda_{2}&0&\\cdots&\\lambda_{n}&0\\\\ 0&\\lambda_{1}&0&\\lambda_{2}&\\cdots&0&\\lambda_{n}\\\\ \\lambda_{1}^{2}&0&\\lambda_{2}^{2}&0&\\cdots&\\lambda_{n}^{2}&0\\\\ \\vdots&\\vdots&\\vdots&\\vdots&\\ddots&\\vdots&\\vdots\\\\ 0&\\lambda_{1}^{n-2}&0&\\lambda_{2}^{n-2}&\\cdots&0&\\lambda_{n}^{n-2}\\\\ \\lambda_{1}^{n-1}&0&\\lambda_{2}^{n-1}&0&\\cdots&\\lambda_{n}^{n-1}&0\\\\ 0&\\lambda_{1}^{n-1}&0&\\lambda_{2}^{n-1}&\\cdots&0&\\lambda_{n}^{n-1} \\end{array} \\right|. \\]
Keywords : Trigonometric system, Wronskian, determinant, linear independence, Abel