Wronski Determinant of Trigonometric System

Year 2022, Volume: 5 Issue: 1, 1 - 8, 01.03.2022

Ufuk Kaya

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

References

• Ahlfors V.L. 1979. Complex Analysis, McGraw-Hill Inc., New York.
• Boyce W.E., DiPrima R.C. 1986. Elementary differential equations and boundary value problems, Wiley, New York.
• Christensen O., Christensen K. 2006. Linear Independence and Series Expansions in Function Spaces, Am. Mat. Mon., 113 (7): 611-627.