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Morera's theorem: vanishing triangle integrals characterize holomorphy among continuous functions

Statement

A continuous function on an open subset of $C$ is holomorphic if and only if its integral around the boundary of every filled triangle contained in the open set is zero.

Precisely, if $Ω \subseteq C$ is open and $f : Ω \to C$ is continuous, then

$f is holomorphic on Ω ⟺ \int_{\partial Δ [a, b, c]} f (z) d z = 0 whenever Δ [a, b, c] \subseteq Ω .$

Repeated or collinear vertices are permitted.

Facts & Assumptions

Given: An open set $Ω \subseteq C$ and a continuous function $f : Ω \to C$ .

[L1]

A filled triangle $Δ [a, b, c]$ has positively oriented boundary $ℓ_{a b} * ℓ_{b c} * ℓ_{c a}$ , and repeated or collinear vertices are allowed (Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter).

[L2]

On an open set star-shaped with respect to a point, a continuous function whose integral vanishes around every contained filled triangle has a holomorphic primitive $F$ satisfying $F^{'} = f$ (Vanishing integrals around triangles construct a primitive for a continuous function on a star-shaped domain).

[L3]

Every holomorphic function has complex derivatives of every natural order locally (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle).

[L4]

A holomorphic function has zero integral around every filled triangle contained in its open domain, including degenerate triangles (Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain).

Proof

technique · direct

1.1

For the vanishing-integrals-to-holomorphy direction, fix $a \in Ω$ and choose $r > 0$ with $\overline{D} (a, r) \subseteq Ω$ ; the disc $D (a, r)$ is star-shaped with respect to $a$ , and every filled triangle in it is among the triangles covered by the assumed condition and [L1].

givenL1

1.2

For the holomorphy-to-vanishing-integrals direction, if $f$ is holomorphic on $Ω$ , [L4] gives zero integral around every filled triangle of [L1] contained in $Ω$ , including those with repeated or collinear vertices.

L1L4

2.1

For the vanishing-integrals-to-holomorphy direction, [L2] applied on $D (a, r)$ supplies a holomorphic function $F$ there with $F^{'} = f$ .

step 1.1L2

3.1

For the vanishing-integrals-to-holomorphy direction, [L3] makes the derivative $F^{'}$ holomorphic, so $f = F^{'}$ is holomorphic on $D (a, r)$ .

step 2.1L3

4.1

If $Ω$ is nonempty, the point in step 1.1 was arbitrary, so step 3.1 proves holomorphy throughout $Ω$ under the integral condition, while step 1.2 proves the converse; if $Ω$ is empty, both directions are vacuous.

step 1.1step 1.2step 3.1∎

Depends on

Used by

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25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §2.3 (standard reference, not scraped)
E. Stein and R. Shakarchi, Complex Analysis, Ch. 2, Theorem 5.1 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Theorem 2.3.4 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, Theorem 2.21 (standard reference, not scraped)