For a real free scalar field theory, the action (up to a total derivative) can be written in quadratic form (with two fields sandwiching the Klein-Gordon operator)  $$ S[\phi]=-\frac{1}{2}\int d^4x \phi(x) (\partial_\mu\partial^\mu+m^2)\phi(x) \nonumber \\=\lim_{\epsilon\rightarrow0}-\frac{1}{2}\int d^4x \phi(x) (\partial_\mu\partial^\mu+m^2-i\epsilon)\phi(x).$$ The generating functional is defined to be the path integral $$ Z[j] = \int [d\phi] e^{iS[\phi]+i\int d^4x j(x)\phi(x)}, $$ where \(j(x)\) is a particle source,similar to chemical potential in statistical mechanics. BTW, Z is similar to the partition function in statistical mechanics. The generating functional for the free theory is usually denoted with an “o” subscript \(Z_o\).

To evaluate this path integral, use the shifted field \(\phi’=\phi+\int d^4y\Delta_F(x-y)j(y)\), where the Feynman propagator \(\Delta_F(x-y)\) is the Green function of the Klein-Gordon operator \(\lim\limits_{\epsilon\rightarrow0}(\partial_\mu\partial^\mu+m^2-i\epsilon)\Delta_F(x-y)=-\delta^{(d)}(x-y) \) $$ S[\phi] = S[\phi’]-\int d^4x \phi'(x)j(x)-\frac{i}{2}\int d^4x\int d^4y j(x)i\Delta_F(x-y)j(y)$$ $$\text{and } \int d^4x j(x)\phi(x)=\int d^4x j(x)\phi'(x)+i\int d^4x \int d^4y j(x)i\Delta_F(x-y)j(y). $$ Therefore the exponent in the generating functional $$ iS[\phi]+i\int j\phi = iS[\phi’]-i\int \phi’j+\frac{1}{2}\iint ji\Delta j+i\int \phi’j-\iint ji\Delta j \\ = iS[\phi’]-\frac{1}{2}\int d^4x \int d^4 y j(x) i\Delta_F(x-y)j(y),$$ and $$ Z_o[j] = Z_o[0] \exp\left(-\frac{1}{2}\int d^4x \int d^4 y j(x) i\Delta_F(x-y)j(y)\right).$$