Morera's theorem states that 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.
