  We derive an exact path integral formulation for the partition function for
the Ising model using a mapping between spins and poles of a Laurent expansion
for a field on the complex plane. The advantage in using this formulation for
the evaluation of the partition function and $n-$point functions are twofold.
First of all, we show that this mapping is independent of the couplings, and
that for $h=0$ it is possible to perform a low temperature expansion as a
perturbation theory via Feynman diagrams. The couplings are mapped naturally to
a propagator for a complex field. The combinatorial nature of the partition
function is shown to lead to an auxiliary field with a non-zero external field
interaction which enforces the spin-like nature. Feynman diagrams are shown to
coincide with certain combinations of traces of coupling inverses in a certain
rescaling.
