Morera,s Theorem
المؤلف:
Arfken, G.
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press
الجزء والصفحة:
...
17-11-2018
1040
Morera's Theorem
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.
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