The problem of missing terms in term by term integration involving divergent integrals

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity condition, the problem of missing terms for the incomplete Stieltjes transform, $\int_0^a f(x) (\omega+x)^{-1} \mathrm{d}x$, and the Stieltjes transform itself, $\int_0^{\infty} f(x) (\omega+x)^{-1} \mathrm{d}x$, is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex valued function $f(z) (\omega + z)^{-1}$, with the divergent integrals arising from term by term integration interpreted as finite part integrals.