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# 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.

Cauchy Integral Theorem, Contour Integration

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## 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.

Morera's Theorem

## Cite this as:

Weisstein, Eric W. "Morera's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MorerasTheorem.html