Explicit half-range expansions for Sturm-Liouville operators

Boundary value problems for linear transport equations sometimes require th e explicit construction of solutions when boundary conditions are prescribe d only on parts of the boundary and thus necessitate the construction of ha lf-range expansions. In contrast to the standard eigenfunction expansion, a half-range expansion of a given function must be reconstructed just over h alf the domain, using just half of the eigenfunctions. The difficulty of su ch expansions arises because the eigenfunctions are not orthogonal, though they are complete, over half the domain, and there is no obvious method of obtaining the expansion coefficients. Here we use complex variable techniqu es to find explicit formulas for the coefficients of half-range expansions for regular, negative definite Sturm-Liouville operators. We prove that the half-range expansion formula is unique, and find the corresponding half-ra nge Green functions.