  We use the residue theorem to derive an expression for the number of lattice
oints in a dilated n-dimensional tetrahedron with vertices at lattice points on
each coordinate axis and the origin. This expression is known as the Ehrhart
polynomial. We show that it is a polynomial in t, where t is the integral
dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these
tetrahedra, relating the Ehrhart polynomials of the interior and the closure of
the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient
for codimension 2. Finally, we show how our ideas can be used to compute the
Ehrhart polynomial for an arbitrary convex lattice polytope.
