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Zbl 0622.33005
Naylor, D.
On a simplified asymptotic formula for the Mathieu function of the third kind. (English)
[J] SIAM J. Math. Anal. 18, 1616-1629 (1987). ISSN 0036-1410; ISSN 1095-7154/e

This paper considers the asymptotic form of solutions of the equation $y\sb{xx}=(u\sp 2-2h\sp 2 \cosh 2x)y$ for fixed real values of x and h and large complex values of u. Attention is focused on that solution known as the Mathieu function of the third kind, $M\sb{\nu}\sp{(3)}(x)$, and for values of u in the half plane Re(u)$\ge 0$. The basic asymptotic formulas require the determination of an elliptic integral but, when u is large, it is shown how this integral can be approximated by elementary functions.

MSC 2000:: *33E10 Spheroidal wave functions, etc.
41A60 Asymptotic problems in approximation

Keywords: Mathieu function

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