IV.2 Ordinary Differential Equations

which, again, we can solve using a Cauchy integral and (31). The problem of finding K± is more delicate. At first sight, we could take the logarithm of (33), giving logK+− logK− = logG. This looks similar to (31), but it usually happens that logG(t) is not continuous for all t ∈ C , which means that we cannot use (30). However, this difficulty can be overcome. The problem of finding K± such that (33) is satisfied is also the key step in the Wiener–Hopf technique (a method for solving linear PDEs with mixed boundary conditions and semi-infinite geometries). In that context, a typical problem would be: factor a given function L(z) as L(z) = L+(z)L−(z), where L+(z) is analytic in an upper half-plane, Imz > a, L−(z) is analytic in a lower half-plane, Imz < b, and a < b so that the two half-planes overlap. There are also related problems where L is a 2 × 2 or 3 × 3 matrix; it is not currently known how to solve such matrix Wiener–Hopf problems except in some special cases.