  The bulk-boundary correspondence relates topologically-protected edge modes
to bulk topological invariants, and is well-understood for short-range
free-fermion chains. Although case studies have considered long-range
Hamiltonians whose couplings decay with a power-law exponent $\alpha$, there
has been no systematic study for a free-fermion symmetry class. We introduce a
technique for solving gapped, translationally invariant models in the 1D BDI
and AIII symmetry classes with $\alpha>1$, linking together the quantized
winding invariant, bulk topological string-order parameters and a complete
solution of the edge modes. The physics of these chains is elucidated by
studying a complex function determined by the couplings of the Hamiltonian: in
contrast to the short-range case where edge modes are associated to roots of
this function, we find that they are now associated to singularities. A
remarkable consequence is that the finite-size splitting of the edge modes
depends on the topological winding number, which can be used as a probe of the
latter. We furthermore generalise these results by (i) identifying a family of
BDI chains with $\alpha<1$ where our results still hold, and (ii) showing that
gapless symmetry-protected topological chains can have topological invariants
and edge modes when $\alpha -1$ exceeds the dynamical critical exponent.
