If  is continuous in a region  and satisfies

for all closed contours  in , then  is analytic in .

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.

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 373-374, 1985.

Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 26, 1999.