Morera’s theorem

Morera’s theorem provides the converseMathworldPlanetmath of Cauchy’s integral theorem.

Theorem [1] Suppose G is a region in , and f:G is a continuous functionMathworldPlanetmathPlanetmath. If for every closed triangle Δ in G, we have


then f is analyticPlanetmathPlanetmath on G. (Here, Δ is the piecewise linear boundary ( of Δ.)

In particular, if for every rectifiable closed curve Γ in G, we have Γf𝑑z=0, then f is analytic on G. Proofs of this can be found most undergraduate books on complex analysis [2, 3].


  • 1 W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Inc., 1987.
  • 2 E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.
  • 3 R.A. Silverman, Introductory Complex Analysis, Dover Publications, 1972.
Title Morera’s theorem
Canonical name MorerasTheorem
Date of creation 2013-03-22 12:58:09
Last modified on 2013-03-22 12:58:09
Owner matte (1858)
Last modified by matte (1858)
Numerical id 12
Author matte (1858)
Entry type Theorem
Classification msc 30D20