Stokes multipliers for a class of ordinary differential equations

Abstract

A new method is presented for calculating the Stokes multipliers for a class of linear second-order ordinary differential equations. The Stokes multipliers allow the asymptotic solutions of these equations to be continued across the Stokes lines on which they are dominant. The differential equations, of the class considered here, have an irregular singular point at infinity and a singular point at the origin, which may be either regular or irregular. The Stokes multipliers, as functions of the coefficients in the differential equation, are obtained in the form of convergent infinite series, whose terms must be obtained from the solution of recursion relations, which are derived. In the case of Whittaker’s equation (when the origin is a regular singular point), the known results are obtained analytically. When the origin is an irregular singular point, numerical evaluation of the series is necessary, but the method seems to be quite efficient for use with digital computers. In the special case of an equation, with two irregular singular points, which can be transformed to Mathieu’s equation, the numerical results for the Stokes multiplier show good agreement with available known results for the characteristic exponents of Mathieu’s equation.