Given the Mathieu equation, $\frac{d^2w}{dz^2}+(a-2q\cos 2z)y=0$, I have the solution
\begin{align} y(t)=A_1C(a,q,z)+A_2S(a,q,z)\ , \end{align} where $C(\cdot,\cdot,\cdot)$ and $S(\cdot,\cdot,\cdot)$ are the Mathieu cosine and sine solutions, respectively. Does an expression exist describing the asymptotics of the solution as $z\to\pm\infty$?
I've tried looking at the Digital Library of Mathematical Functions, but they only have the asymptotic expansions for small and large $q$. I know that one can let $z\to\pm iz'$ to find the modified Mathieu functions. An asymptotic series exists for this case, but it is only for $\Re z'\to\infty$, so I don't think it is valid to take the asymptotic form of the modified Mathieu equation and let $z'\to\pm z$ as this would correspond to $\Re z'\to\pm i\infty$.
I've tried looking at academic papers, but they mainly deal with asymptotics of $q$, rather than $z$. This makes me wonder, is the asymptotic form of the Mathieu cosine and sine functions known for $z\to\pm\infty$?
