proof of Morera’s theorem

We provide a proof of Morera’s theoremMathworldPlanetmath under the hypothesisMathworldPlanetmathPlanetmath that Γf(z)𝑑z=0 for any circuitMathworldPlanetmath Γ contained in G. This is apparently more restrictive, but actually equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, to supposing Δf(z)𝑑z=0 for any triangleMathworldPlanetmath ΔG, provided that f is continuousPlanetmathPlanetmath in G.

The idea is to prove that f has an antiderivative F in G. Then F, being holomorphic in G, will have derivativesPlanetmathPlanetmath of any order in G; but F(n)(z)=f(n-1)(z) for all zG, n, n1.

First, suppose G is connectedPlanetmathPlanetmath. Then G, being open, is also pathwise connected.

Fix z0G. For any zG define F(z) as

F(z)=γ(z0,z)f(w)𝑑w, (1)

where γ(z0,z) is a path entirely contained in G with initial point z0 and final point z.

The function F:G is well defined. In fact, let γ1 and γ2 be any two paths entirely contained in G with initial point z0 and final point z; define a circuit Γ by joining γ1 and -γ2, the path obtained from γ2 by “reversing the parameter direction”. Then by linearity and additivity of integral

Γf(w)𝑑w=γ1f(w)𝑑w+-γ2f(w)𝑑w=γ1f(w)𝑑w-γ2f(w)𝑑w; (2)

but the left-hand side is 0 by hypothesis, thus the two integrals on the right-hand side are equal.

We must now prove that F=f in G. Given zG, there exists r>0 such that the ball Br(z) of radius r centered in z is contained in G. Suppose 0<|Δz|<r: then we can choose as a path from z to z+Δz the segment γ:[0,1]G parameterized by tz+tΔz. Write f=u+iv with u,v:G: by additivity of integral and the mean value theorem,

F(z+Δz)-F(z)Δz = 1Δzγf(w)𝑑w
= 1Δz01f(z+tΔz)Δz𝑑t
= u(z+θuΔz)+iv(z+θvΔz)

for some θu,θv(0,1). Since f is continuous, so are u and v, and


In the general case, we just repeat the procedure once for each connected componentMathworldPlanetmathPlanetmath of G.

Title proof of Morera’s theorem
Canonical name ProofOfMorerasTheorem
Date of creation 2013-03-22 18:53:34
Last modified on 2013-03-22 18:53:34
Owner Ziosilvio (18733)
Last modified by Ziosilvio (18733)
Numerical id 10
Author Ziosilvio (18733)
Entry type Proof
Classification msc 30D20