We examine the zero momentum limit of the finite temperature vacuum polarization for a quantized scalar field coupled to a classical external field. Ordinarily, this type of Feynman diagram is plagued by nonanalytic behaviour when the external momentum tends to zero. Using imaginary-time, we show that this behaviour is not present in an exact background field solution and how the Feynman diagram calculation may be trivially modified to match the exact solution. Comparisons with recent real-time results are also made.