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Zbl 0557.33005
Naylor, D.
On simplified asymptotic formulas for a class of Mathieu functions. (English)
[J] SIAM J. Math. Anal. 15, 1205-1213 (1984). ISSN 0036-1410; ISSN 1095-7154/e

Author's abstract. This paper considers the asymptotic form of solutions of the equation $y\sb{xx}=(u\sp 2+2h\sp 2 \cosh 2x)y$ for real values of x and h and large values of u. Attention is focussed on the solution $\psi$ (x,u) that tends to zero as $x\to \infty$ and for values of u in the half plane Re(u)$\ge 0$. The basic asymptotic formulas that appear require the determination of an elliptic integral but, when u is large, it is shown how this integral can be suitably approximated by elementary functions. An asymptotic formula is derived which gives the large zeros of the function $\psi$ (x,u) regarded as a function of u, the quantity x being supposed prescribed and positive.
[P.A.McCoy]

MSC 2000:: *33E10 Spheroidal wave functions, etc.

Keywords: Mathieu functions; elliptic integral; asymptotic formula

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