Stokes constants for a certain class of second-order ordinary differential equations

Abstract

By generalizing the coupled wave integral equations method devised by Hinton, the Stokes phenomena of the standard second-order ordinary differential equations with the following coefficient functions q(z) are analyzed: (i) q(z)=a2Nz2N+∑j=−∞N−1ajzj; (ii) q(z)=a2N−1z2N−1+∑j= −∞N−2ajz j; (iii) q(z)=∑j=04ajz j, with a3=0; and (iv) q(z)=∑j=03ajz j, with a2=0. First the relations are derived among the Stokes constants, which enable the expression of all Stokes constants in terms of only one. Furthermore, this one Stokes constant is shown to be expressed in the analytical form of a convergent infinite series as a function of the coefficients of q(z). This was done for all four cases listed above.