Integral representation of the Ising model

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The partition function of the 2D Ising model coupled to an external magnetic field is studied. We show that the sum over the spin variables can be reduced to an integration over a finite number of variables. This integration must be performed numerically. But in order to reduce the partition function we must introduce as many different coupling constants as spin variables. The total memory that we need in order to store these coupling constants imposed important restrictions on the number of spin variables.