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Morera's Theorem


If f(z) is continuous in a region D and satisfies

 ∮_gammafdz=0

for all closed contours gamma in D, then f(z) is analytic in D.

Morera's theorem does not require simple connectedness, which can be seen from the following proof. Let D be a region, with f(z) continuous on D, and let its integrals around closed loops be zero. Pick any point z_0 in D, and pick a neighborhood of z_0. Construct an integral of f,

 F(z)=int_(z_0)^zf(z)dz.

Then one can show that F^'(z)=f(z), and hence F is analytic and has derivatives of all orders, as does f, so f is analytic at z_0. This is true for arbitrary z_0 in D, so f is analytic in D.

It is, in fact, sufficient to require that the integrals of f around triangles be zero, but this is a technical point. In this case, the proof is identical except F(z) must be constructed by integrating along the line segment z_0z^_ instead of along an arbitrary path.


See also

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.

Referenced on Wolfram|Alpha

Morera's Theorem

Cite this as:

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

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