Apostol uses the Abel Identity developed early in his book as Theorem 4.2 (image below)
$$ \sum_{y<n\leq x}= A(x)f(x) - A(y)f(y) - \int_{y}^{x}A(t)f'(t) dt $$
to prove a result about complex Dirichlet series (11.6 Lemma 2, image below).
Question - Why is the Abel Identity valid in the complex domain?
My thoughts
The derivation he presents for Theorem 4.2 is only for functions in the real domain.
