Morera's Theorem

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


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,


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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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.

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