# Analytic Solutions of the Eigenvalues of Mathieu’s Equation

*Chein-Shan Liu*

*Journal of Mathematics Research*, 2020, vol. 12, issue 1, 1

**Abstract:**
Mathieu’s eigenvalue problem −y′′(x) + 2e_0 cos(2x)y(x) = λy(x), 0 < x < ℓ is symmetric if cos(2x) = cos(2ℓ − 2x) for ℓ = k0π, k0 ∈ N, and skew-symmetric if cos(2x) = − cos(2ℓ − 2x) for ℓ = π/2. Two typical boundary conditions are considered. When the eigenfunctions are expanded by the orthonormal bases of sine functions or cosine functions, we can derive an n-dimensional matrix eigenvalue problem, endowing with a special structure of the symmetric coefficient matrix A -= [a_ij], a_ij = 0 if i + j is an odd integer. Based on it, we can obtain the eigenvalues easily and analytically. When ℓ = k_0π, k_0 ∈ N, we have a_ij = 0 if |i − j| > 2k_0. Besides the diagonal band, A has two off-diagonal bands, and furthermore, a cross band appears when k_0 ≥ 2. The product formula, the recursion formulas of characteristic functions and a fictitious time integration method (FTIM) are developed to find the eigenvalues of Mathieu’s equation.

**JEL-codes:** R00 Z0 (search for similar items in EconPapers)

**Date:** 2020

**References:** View complete reference list from CitEc

**Citations:** Track citations by RSS feed

**Downloads:** (external link)

http://www.ccsenet.org/journal/index.php/jmr/article/download/0/0/41554/43133 (application/pdf)

http://www.ccsenet.org/journal/index.php/jmr/article/view/0/41554 (text/html)

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:ibn:jmrjnl:v:12:y:2020:i:1:p:1

Access Statistics for this article

More articles in Journal of Mathematics Research from Canadian Center of Science and Education Contact information at EDIRC.

Bibliographic data for series maintained by Canadian Center of Science and Education ().