# Asymptotics of Mathieu functions

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 $$z\to\pm\infty$$?

• Have you tried using WKB Anstaz for deriving relations that can lead to asymptotics? Aug 9, 2019 at 7:37
• I thought that WKB would only be useful for the limiting cases of large/small $a$ and $q$. I'll need to try it. Aug 10, 2019 at 20:25