Correction:

I wrote (regarding choosing a domain D in C on which log is defined and continuous):

<<
More generally, D can be chosen to be any simply-connected region of the Riemann surface for log. 
>>

This should read instead:

More generally, D can be chosen to be any bounded open subset of the Riemann surface for log.

P.S. The 1-dimensional product integral can be adapted to work over curves in C as well.

--Dan