  The Mathieu operator {equation*} L(y)=-y"+2a \cos{(2x)}y, \quad a\in
\mathbb{C},\;a\neq 0, {equation*} considered with periodic or anti-periodic
boundary conditions has, close to $n^2$ for large enough $n$, two periodic (if
$n$ is even) or anti-periodic (if $n$ is odd) eigenvalues $\lambda_n^-$,
$\lambda_n^+$. For fixed $a$, we show that {equation*} \lambda_n^+ -
\lambda_n^-= \pm \frac{8(a/4)^n}{[(n-1)!]^2} [1 - \frac{a^2}{4n^3}+ O
(\frac{1}{n^4})], \quad n\rightarrow\infty. {equation*} This result extends the
asymptotic formula of Harrell-Avron-Simon, by providing more asymptotic terms.
