the asymptotic form of eigenvalues for a class of sturm-liouville problem with one simple turning point

the purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of sturm-liouville problem with equation of the form w??=(?2f(x)?r(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in c2(a,b which has a single zero (so called turning point) at point 0x=x and r(x) is a continuously differentiable function. we prove that, as a classical case, the asymptotic form of eigenvalues of (1) with periodic boundary condition w(a)=w(b), as well as with semi-periodic boundary condition w?(a)=w?(b)w(a)=?w(b), are the same as dirichlet boundary condition w?(a)=?w?(b)w(a)=0=w(b). we also study the asymptotic formula for the eigenvalues of (1) with boundary condition w?(a)=0=w(b), as well as w(a)=0=w?(b) and w?(a)=0=w(b).