  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.
