Connection Formulas for Second-Order Differential Equations Having an Arbitrary Number of Turning Points of Arbitrary Multiplicities

Consider the differential equation \[d^2 {w / {dx^2 }} = \left\{ {u^2 f(x) + g(x)} \right\}w,\quad ,x \in (a,b),\] in which $(a,b)$ is a finite or infinite open interval, u is a positive parameter, $f(x)$ is real and twice continuously differentiable, and $g(x)$ is continuous. It is well known that in any subinterval of $(a,b)$ not containing a turning point, that is, a zero of $f(x)$, uniform asymptotic solutions for large u can be constructed in terms of the so-called Liouville–Green or WKBJ functions: \[f^{ - {1 / 4}} (x)\exp \left\{ { \pm u\int {f^{ - {1 / 2}} (x)dx} } \right\}.\] If $(a,b)$ contains turning points, then differing combinations of the Liouville–Green functions have to be used in subintervals that are separated by a turning point in order to represent the same solution.This paper solves the general problem of connecting the Liouville–Green approximations throughout the interval $(a,b)$ for any number of turning points of arbitrary multiplicities. Several illustrative examples are given,...