This paper deals with Hamiltonians of the form $H=-{\bf \nabla}^2+v(\rr)$,
with $v(\rr)$ periodic along the $z$ direction, $v(x,y,z+b)=v(x,y,z)$. The
wavefunctions of $H$ are the well known Bloch functions
$\psi_{n,\lambda}(\rr)$, with the fundamental property
$\psi_{n,\lambda}(x,y,z+b)=\lambda \psi_{n,\lambda}(x,y,z)$ and
$\partial_z\psi_{n,\lambda}(x,y,z+b)=\lambda
\partial_z\psi_{n,\lambda}(x,y,z)$. We give the generic analytic structure
(i.e. the Riemann surface) of $\psi_{n,\lambda}(\rr)$ and their corresponding
energy, $E_n(\lambda)$, as functions of $\lambda$. We show that $E_n(\lambda)$
and $\psi_{n,\lambda}(x,y,z)$ are different branches of two multi-valued
analytic functions, $E(\lambda)$ and $\psi_\lambda(x,y,z)$, with an essential
singularity at $\lambda=0$ and additional branch points, which are generically
of order 1 and 3, respectively. We show where these branch points come from,
how they move when we change the potential and how to estimate their location.
Based on these results, we give two applications: a compact expression of the
Green's function and a discussion of the asymptotic behavior of the density
matrix for insulating molecular chains.