  By the appropriate use of the Fock-Schwinger gauge properties, we derive the
closed integral form of the `point-split' non-local background gauge connection
originally expressed as a finite sum. This is achieved in the limit when the
finite sum becomes infinite. With this closed integral form of the connection,
we obtain the same exact results in the calculation of one-loop effective
Lagrangian accommodating arbitrary orders of covariant field derivatives in
quantum field theory of arbitrary spacetime dimensions and of arbitrary gauge
group. Particularly, we display the one-loop effective Lagrangian for real
boson fields up to 8 mass dimensions-the same result obtained when the
connection was yet in the finite sum form.
