Morera's theorem does not require simple connectedness, which can be seen from the following proof. Let
be a region, with
continuous on ,
and let its integrals around closed loops be zero. Pick any point , and pick a neighborhood
of .
Construct an integral of ,

Then one can show that ,
and hence
is analytic and has derivatives of all orders,
as does ,
so
is analytic at .
This is true for arbitrary , so is analytic in .

It is, in fact, sufficient to require that the integrals of around triangles be zero, but this is a technical point. In
this case, the proof is identical except must be constructed by integrating along the line segment
instead of along an arbitrary path.