  We review and present full detail of the Feynman diagram - based and
heat-kernel method - based calculations of the simplest nonlocal form factors
in the one-loop contributions of a massive scalar field. The paper has a
pedagogical and introductory purposes and is intended to help the reader in
better understanding the existing literature on the subject. The functional
calculations are based on the solution by Avramidi and Barvinsky \& Vilkovisky
for the heat kernel and are performed in curved spacetime. One of the important
points is that the main structure of non-localities is the same as in the flat
background.
