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

Consider the differential equation \[d^2 {w / {dx^2 }} = \{ {u^2 f(x) + g(x)} \}w, ,x ∈ (a,b),\] 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: \[f^{ - {1 / 4}} (x)exp \{ { ± u∫ {f^{ - {1 / 2}} (x)dx} } \}.\] 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, including an arbitrary number of turning points of even multiplicity, an arbitrary number of turning points of odd multiplicity, an eigenvalue problem involving four turning points of multiplicities 1, 2, 3, and 4, and a problem with two simple turning points and one multiple turning point that is solvable exactly-in terms of Whittaker functions.