Author’s abstract. This paper considers the asymptotic form of solutions of the equation

${y}_{xx}=({u}^{2}+2{h}^{2}cosh2x)y$ 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.