In this note we reveal that the missing link among a few crucial results in
analysis, Abel continuity theorem, convergence theorem on (generalized) Dirichlet
series, Paley-Wiener theorem is the Laplace transform with Stieltjes integration. By
this discovery, the reason why the domains of Stoltz path and of convergence look
similar is made clear. Also as a natural intrinsic property of Stieltjes integral, the use
of partial summation in existing proofs is elucidated. Secondly, we shall reveal that a
basic part of the proof of Paley-Wiener theorem is a version of the Laplace transform.
Keywords: Laplace transform, Stieltjes integral, Abel continuity theorem, Paley-Wiener theorem, conformal mapping, 2010 MSC: 130E99, 44A10, 40A05.
