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Morera's theorem

Morera's theorem is a fundamental result in complex analysis named after the Italian mathematician and engineer Giacinto Morera, who established it in 1886.^[1] It provides a sufficient condition for a complex-valued function to be holomorphic (analytic) in a domain, serving as a partial converse to Cauchy's integral theorem.^[2] Specifically, the theorem states that if a function f: \Omega \to \mathbb{C} is continuous on a region \Omega \subseteq \mathbb{C} (an open connected set) and satisfies \int_\gamma f(z) \, dz = 0 for every simple closed rectifiable curve \gamma in \Omega, then f is analytic throughout \Omega.^[2] Unlike Cauchy's integral theorem, which assumes analyticity to conclude that contour integrals over closed paths vanish, Morera's theorem reverses this implication under the additional hypothesis of continuity, thereby characterizing analytic functions via their integral properties.^[3] A common practical formulation requires the integral condition to hold only over all closed triangular contours in \Omega, which suffices due to the geometry of regions in the complex plane and Goursat's theorem.^[3] This version is especially useful in applications, as verifying integrals over triangles is often more straightforward than over arbitrary contours.^[4] The proof of Morera's theorem typically proceeds by constructing an antiderivative (primitive) F(z) = \int_{z_0}^z f(\zeta) \, d\zeta for a fixed z_0 \in \Omega, demonstrating that F'(z) = f(z) everywhere in \Omega, and thus f inherits analyticity from F.^[2] The theorem highlights the deep connection between differentiability and integrability in complex analysis, enabling proofs of analyticity for functions like power series expansions or integrals without explicit differentiation.^[4] Notably, the result holds without requiring \Omega to be simply connected, distinguishing it from some related integral theorems.^[3]

Statement and History

Formal Statement

Morera's theorem provides an integral condition sufficient for a continuous function to be holomorphic in a domain of the complex plane. Specifically, let D be an open connected set in \mathbb{C}, and let f: D \to \mathbb{C} be a continuous function. If

\int_{\gamma} f(\zeta) \, d\zeta = 0

for every closed piecewise C^1 curve \gamma in D, then f is holomorphic on D.^[3]^[2] An equivalent formulation states that under these hypotheses, f admits a primitive (antiderivative) F: D \to \mathbb{C} on D, meaning F is holomorphic on D and F' = f.^[4]^[5]

Historical Development

Morera's theorem was proved by the Italian mathematician and engineer Giacinto Morera in 1886 as a converse to Cauchy's integral theorem, which states that the integral of a holomorphic function over a closed path vanishes.^[6] This result provided a criterion for verifying holomorphy through the vanishing of contour integrals, reversing the implication of Cauchy's theorem under suitable continuity conditions.^[7] The theorem first appeared in Morera's paper titled "Un teorema fondamentale nella teoria di funzioni di variabile complessa," published in the Rendiconti del Reale Istituto Lombardo di Scienze e Lettere (volume 19, pages 304–308).^[8] Morera, who had studied under prominent figures like Eugenio Beltrami and Enrico Betti, contributed this work amid his broader research on complex variables and potential theory during his tenure at the University of Genoa.^[9] The theorem emerged in the late 19th century, a pivotal era for complex analysis following Cauchy's early 19th-century advancements in path integrals and Riemann's 1850s developments of the Riemann integral and the notion of analytic continuation.^[7] Morera's proof underscored the interplay between continuity of the function and the zero-integral condition along closed paths, addressing gaps in understanding holomorphy beyond purely algebraic contexts.^[6] Formally known as Morera's theorem since its publication, it facilitated subsequent explorations of analytic properties and paved the way for proofs of holomorphy in non-algebraic settings, influencing later works such as those refining criteria for differentiability in complex domains.^[9]

Theoretical Foundations

Relation to Cauchy's Theorem

Cauchy's theorem asserts that if a function f is holomorphic on a simply connected domain D \subseteq \mathbb{C}, then for any closed contractible curve \gamma in D, the line integral \int_\gamma f(z) \, dz = 0.^[10] This result forms a cornerstone of complex analysis, linking differentiability to integral properties over contractible paths.^[10] Morera's theorem serves as a partial converse to Cauchy's theorem, establishing the reverse implication under an additional continuity hypothesis: if f: D \to \mathbb{C} is continuous on a domain D and \int_T f(z) \, dz = 0 for every closed triangular path T in D, then f is holomorphic on D.^[2] In simply connected domains, where every closed curve is contractible to a point, this condition on triangles suffices to capture all relevant integrals, yielding an equivalence between holomorphicity and the vanishing of integrals over closed paths for continuous functions.^[2] Thus, the two theorems together provide an integral characterization of holomorphic functions within this setting. In more general domains that are not simply connected, such as the punctured plane \mathbb{C} \setminus \{0\}, the interplay requires refinements involving homology or winding numbers, as holomorphic functions like f(z) = 1/z exhibit nonzero integrals over non-contractible loops despite satisfying Cauchy's theorem locally on contractible subdomains.^[2] Morera's theorem remains local in nature, relying on triangles to ensure differentiability pointwise, but the global integral behavior in multiply connected regions demands additional topological considerations to fully align with Cauchy's theorem.^[10] The continuity assumption in Morera's theorem is essential, as its absence can lead to functions satisfying the integral condition yet failing to be holomorphic. A classic counterexample is the function

f(z) = \begin{cases} \frac{1}{z^2} & \text{if } z \neq 0, \\ 0 & \text{if } z = 0, \end{cases}

defined on \mathbb{C}. For any closed triangular path \gamma in \mathbb{C} not passing through z=0, \int_\gamma f(z) \, dz = 0, since the residue of $1/z^2 at z=0 is zero and triangles not enclosing zero yield zero by Cauchy's theorem on the punctured plane. However, f is discontinuous at z=0 and thus not holomorphic there.^[11]

Key Prerequisites

A holomorphic function is a complex-valued function that is complex differentiable at every point in its domain, meaning the limit defining the derivative exists and is the same regardless of the direction of approach in the complex plane.^[12] Such functions can be expressed in terms of their real and imaginary parts satisfying the Cauchy-Riemann equations, which provide necessary conditions for differentiability in the complex sense.^[13] In complex analysis, continuity refers to the property that small changes in the input produce small changes in the output, formalized by the limit of the function equaling its value at the point as the input approaches that point.^[14] While continuity is a weaker condition than differentiability, it ensures that functions are integrable along paths, allowing line integrals to be well-defined; however, mere continuity does not imply the stronger properties associated with holomorphic functions, such as the existence of primitives locally.^[14] Line integrals in the complex plane are defined for a function f(z) along a path \gamma, parameterized by a piecewise continuously differentiable curve, as \int_\gamma f(z) \, dz = \int_a^b f(\gamma(t)) \gamma'(t) \, dt, where the integral on the right is a standard real integral.^[15] These integrals measure the accumulation of the function along the path and are fundamental for studying properties like path independence, which relies on the continuity of f to guarantee the integral's existence.^[15] The domain D in complex analysis must be open to allow for neighborhoods around each point where local properties like differentiability hold, and connected to ensure global consistency, such as path-connectedness that permits the construction of primitives for holomorphic functions across the entire domain.^[16] Without openness, boundary points could disrupt differentiability, and without connectedness, the domain might split into components where integrals behave independently, preventing unified theorems on primitives.^[17]

Proof

Core Construction

To construct an antiderivative for the continuous function f on the domain D, fix a point z_0 \in D. Since D is connected, for any z \in D there exists a piecewise smooth path \gamma_z from z_0 to z lying entirely in D. Define

F(z) = \int_{\gamma_z} f(\zeta) \, d\zeta.

^[18]^[19] This definition requires showing that F(z) is independent of the choice of path \gamma_z. Suppose \gamma_1 and \gamma_2 are two such paths from z_0 to z. The difference \int_{\gamma_1} f(\zeta) \, d\zeta - \int_{\gamma_2} f(\zeta) \, d\zeta equals the integral of f over the closed curve formed by traversing \gamma_1 and then -\gamma_2, which vanishes by the hypothesis of Morera's theorem. Thus, F(z) is well-defined regardless of the path taken.^[18]^[19] The connectedness of D guarantees the existence of paths between any pair of points, enabling the global definition of F. If D is simply connected, the construction proceeds directly as above, leveraging the zero-integral condition over all closed curves. For the general connected open domain, the proof extends by verifying the condition on triangular contours via triangulation of regions between paths, ensuring path independence holds locally and extends globally.^[18]

Verification Steps

To verify that the constructed antiderivative F satisfies F'(z) = f(z) for all z in the domain, consider the difference quotient at a point z \in G:

F'(z) = \lim_{h \to 0} \frac{F(z + h) - F(z)}{h}.

By the definition of F from the core construction, F(z + h) - F(z) = \int_{\gamma_h} f(\zeta) \, d\zeta, where \gamma_h is the straight-line path from z to z + h. Thus, the difference quotient becomes

\frac{F(z + h) - F(z)}{h} = \frac{1}{h} \int_{\gamma_h} f(\zeta) \, d\zeta. \tag{1}

To evaluate the limit as h \to 0, parameterize the path \gamma_h by \zeta(t) = z + t h for t \in [0, 1], so d\zeta = h \, dt. Substituting into the integral yields

\int_{\gamma_h} f(\zeta) \, d\zeta = \int_0^1 f(z + t h) h \, dt = h \int_0^1 f(z + t h) \, dt.

Dividing by h gives

\frac{1}{h} \int_{\gamma_h} f(\zeta) \, d\zeta = \int_0^1 f(z + t h) \, dt. \tag{2}

Since f is continuous at z, for every \varepsilon > 0, there exists \delta > 0 such that if |h| < \delta, then |f(z + t h) - f(z)| < \varepsilon for all t \in [0, 1], as |t h| \leq |h| < \delta. Therefore,

\left| \int_0^1 f(z + t h) \, dt - f(z) \right| = \left| \int_0^1 [f(z + t h) - f(z)] \, dt \right| \leq \int_0^1 |f(z + t h) - f(z)| \, dt < \varepsilon.

Taking the limit as h \to 0 shows that \int_0^1 f(z + t h) \, dt \to f(z), so F'(z) = f(z). As this holds for every z \in G and F is differentiable everywhere in G, f = F' is holomorphic on G. This completes the proof of Morera's theorem by establishing the existence of a holomorphic antiderivative for f.

Applications

Uniform Convergence

One of the key applications of Morera's theorem is in establishing the preservation of holomorphy under uniform convergence on compact subsets. Consider a domain D \subset \mathbb{C} and a sequence of functions \{f_n\} that are holomorphic on D, converging uniformly to a function f on every compact subset of D. Then f is continuous on D, and for any closed triangular contour \gamma in D, the integral \int_\gamma f \, dz = 0. By Morera's theorem, f is holomorphic on D.^[20] The proof relies on the properties of uniform convergence. Since each f_n is holomorphic, \int_\gamma f_n \, dz = 0 for any such \gamma. Uniform convergence on the compact set \gamma ensures that

\int_\gamma f \, dz = \lim_{n \to \infty} \int_\gamma f_n \, dz = 0,

as the integral is a continuous linear functional with respect to the uniform norm on \gamma. Uniform convergence on compact subsets also implies the continuity of f on D, satisfying the hypotheses of Morera's theorem.^[21] A classic illustration arises in the theory of power series. For a power series \sum_{n=0}^\infty a_n (z - z_0)^n with radius of convergence R > 0, the partial sums s_n(z) = \sum_{k=0}^n a_k (z - z_0)^k are holomorphic polynomials on the disk D = \{ z : |z - z_0| < R \}, converging pointwise to a holomorphic function f inside D. Moreover, on any compact subset K \subset D with \max_{z \in K} |z - z_0| = r < R, the Weierstrass M-test applies: |a_n (z - z_0)^n| \leq M_n = |a_n| r^n for z \in K, and \sum M_n < \infty since r < R, yielding uniform convergence of \{s_n\} to f on K. Thus, f is holomorphic on D.^[22] This result has profound implications in complex analysis. It enables analytic continuation by allowing the uniform extension of holomorphic functions across overlapping domains while preserving holomorphy. Furthermore, it ensures that the space H(D) of holomorphic functions on D, equipped with the Fréchet topology of uniform convergence on compact subsets, is complete, facilitating the study of infinite series, products, and operator theory in this setting.^[23]

Infinite Series and Integrals

Morera's theorem plays a crucial role in establishing the holomorphy of functions defined by infinite series of holomorphic terms, particularly when uniform convergence on compact subsets allows the limit to inherit the property of having vanishing integrals over closed contours. Consider the Riemann zeta function, defined initially for complex variables s with \operatorname{Re}(s) > 1 by the Dirichlet series

\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.

Each term $1/n^s = \exp(-s \log n) is an entire holomorphic function of s, as it is a composition of holomorphic functions. On any compact subset K of the half-plane \operatorname{Re}(s) > 1, the Weierstrass M-test applies with majorants M_n = 1/n^{\sigma_0} for \sigma_0 > 1 bounding K, ensuring uniform convergence of the series on K. The partial sums are holomorphic, and their uniform limit \zeta(s) satisfies the integral condition of Morera's theorem on K, hence \zeta(s) is holomorphic on \operatorname{Re}(s) > 1.^[24] This technique extends to other special functions represented by series. For instance, the exponential integral \operatorname{Ei}(z) = -\int_{-z}^\infty \frac{e^{-t}}{t} \, dt (principal value for real z > 0) can be expanded as a series \operatorname{Ei}(z) = \gamma + \log z + \sum_{n=1}^\infty \frac{z^n}{n \cdot n!} for z \neq 0, where \gamma is the Euler-Mascheroni constant. Uniform convergence on compacta away from the negative real axis follows from the M-test, and applying Morera's theorem to the partial sums confirms holomorphy in \mathbb{C} \setminus (-\infty, 0]. For infinite integrals, Morera's theorem verifies holomorphy by interchanging integration orders over contours. The Gamma function is defined by

\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x} \, dx

for \operatorname{Re}(\alpha) > 0. Each integrand x^{\alpha-1} e^{-x} is holomorphic in \alpha for fixed x > 0, and the integral converges absolutely. To apply Morera's theorem on a compact subset of \operatorname{Re}(\alpha) > 0, consider a closed triangular contour \gamma in the \alpha-plane. The line integral \int_\gamma \Gamma(\alpha) \, d\alpha = 0 follows from Fubini's theorem, which justifies swapping the order:

\int_\gamma \Gamma(\alpha) \, d\alpha = \int_0^\infty x^{-1} e^{-x} \left( \int_\gamma x^\alpha \, d\alpha \right) dx = 0,

since \int_\gamma x^\alpha \, d\alpha = 0 for each fixed x > 0 by Cauchy's theorem, as x^\alpha = \exp(\alpha \log x) is holomorphic in \alpha. Thus, \Gamma(\alpha) is holomorphic on \operatorname{Re}(\alpha) > 0.

Extensions

Weakened Hypotheses

A weakened form of Morera's theorem restricts the integral condition to closed triangular paths while retaining the continuity assumption on the function. Specifically, if f: D \to \mathbb{C} is continuous on a domain D \subset \mathbb{C} and satisfies \oint_{\partial T} f(z) \, dz = 0 for every closed triangle T (including its interior) contained in D, then f is holomorphic on D.^[25] This version simplifies the verification process compared to the standard theorem, as it avoids checking integrals over arbitrary piecewise smooth closed curves. The proof proceeds by extending the condition from triangles to more general paths. Any polygonal closed path in D can be subdivided into triangles by drawing diagonals from a fixed interior point or by triangulation of the enclosed region, ensuring that the integral over the polygonal path vanishes as a sum (or alternating sum) of integrals over these triangles. Path independence then follows for simple polygonal paths, allowing the construction of a primitive function whose derivative recovers f, thus establishing holomorphy.^[25] In the complex plane, this triangular condition is equivalent to the full Morera condition, since Jordan curves can be approximated by polygonal paths, and the integrals converge under continuity. This weakened hypothesis offers practical advantages in applications, such as proving holomorphy for functions defined via series or integrals, where triangular domains are easier to handle than general contours. However, continuity remains essential; dropping it leads to counterexamples where integrals over triangles vanish but the function is not holomorphic. For instance, define f(z) = 1/z^2 for z \neq 0 and f(0) = 0 on \mathbb{C}; the integrals over any closed triangle are zero by the residue theorem (or Cauchy's theorem where applicable), yet f is discontinuous at the origin and hence not holomorphic there.^[11]

Goursat's Variant

Goursat's theorem, a key extension of Morera's theorem, states that if a function f defined on an open domain D \subset \mathbb{C} satisfies \oint_T f(z) \, dz = 0 for every closed triangular path T (including its interior) contained in D, then f is holomorphic throughout D. This formulation eliminates the need for an a priori continuity assumption on f, as holomorphy implies continuity as a consequence.^[3] This result represents a significant refinement introduced by the French mathematician Édouard Goursat in 1884, which predates and influences the approach in Morera's 1886 contribution.^[26] Unlike the original Morera's theorem, which relies on continuity of f to construct a primitive function and verify differentiability via the fundamental theorem of calculus, Goursat's proof removes this hypothesis through a direct estimation technique on subdivided triangles. The approach fixes a point z_0 \in D and considers small triangles around it, using the zero-integral condition to bound the integrals over the sides of these triangles and demonstrate that the difference quotient [F(z) - F(z_0)] / (z - z_0) (where F is a primitive constructed via path integrals) converges to f(z_0) as z \to z_0, thereby proving local differentiability without invoking continuity. This subdivision method ensures the estimates hold uniformly in small neighborhoods, establishing global holomorphy in D. The broader implications of Goursat's variant lie in its role as a more robust converse to Cauchy's integral theorem, applicable in contexts where continuity is challenging to establish independently, such as proofs of the existence and holomorphy of implicit functions defined by integral equations or series expansions.^[27]

References

[1]
Morera's theorem
Morera, Giacinto (1886), "Un teorema fondamentale nella teorica delle funzioni di una variabile complessa", Rendiconti del Reale Instituto Lombardo di ...
[2]
[PDF] MORERA'S THEOREM Let Ω be a region. Recall some ideas
However, the full converse of Morera's theorem is not true, the function f(z)=1/z on Ω = C − {0} being the standard counterexample. Although f is analytic on Ω,.
[3]
Morera's Theorem -- from Wolfram MathWorld
Complex Analysis · Contours. Morera's Theorem. If f(z) is continuous in a region ... . Morera's theorem does not require simple connectedness, which can be ...
[4]
[PDF] Morera's Theorem - Trinity University
Morera's theorem implies that H is analytic on Ω. To compute H′, fix z0 ∈ Ω and choose r > 0 so that the circle Cr = {|z − z0| = r} is contained in Ω.
[5]
[PDF] 2 Cauchy's Theorem and Its - Princeton University
Morera's theorem, which gives a simple integral characterization ... By the proof of Theorem 2.1 the function f has a primitive F in D that satisfies F0 = f.
[6]
[PDF] Introductory Complex Analysis
There is a converse to this statement, known as Morera's Theorem: If f : U → C is continuous, and R∂∆ f(z)dz = 0 for all closed triangles ∆ ⊂ U, then f ...
[7]
Morera theorem - Encyclopedia of Mathematics
Jan 12, 2014 · A fundamental theorem in complex analysis first proved by G. Morera in [Mo], which is an (incomplete) converse of the Cauchy integral theorem.Missing: Giovanni | Show results with:Giovanni
[8]
Introduction | SpringerLink
Mar 13, 2013 · It is in this spirit we have used the nineteenth century term “complex function”, for example, where a modern mathematician would use “analytic ...
[9]
Publications of Giacinto Morera - MacTutor History of Mathematics
G Morera, Sulla separazione di variabili nelle equazioni del moto di un punto materiale su una superficie, Atti Acc. Sci. Torino XVI (1881), 276-295.Missing: Reale Accademia Scienze
[10]
Giacinto Morera (1856 - 1907) - Biography - MacTutor
Morera studied the fundamental problems which arise in dynamics with particular regard to the use of the Pfaff method applied to Jacobian systems of partial ...Missing: Morera's theorem 1886
[11]
[PDF] Lecture Note for Math 220A Complex Analysis of One Variable
Viewing a point (x, y) in xy-plane, we introduce a corresponding complex number. (1.1) z =: x + iy, C = {z =: x + iy : x, y ∈ IR}.
[12]
Counterexample around Morera's theorem
Apr 3, 2016 · Does the conclusion of Morera's theorem still hold if f is supposed to be continuous only in Ω∖{p}? The answer is negative and we provide a ...
[13]
Holomorphic Function -- from Wolfram MathWorld
A synonym for analytic function, regular function, differentiable function, complex differentiable function, and holomorphic map.
[14]
https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/02%3A_Analytic_Functions/2.03%3A_Limits_and_Continuous_Functions
[15]
[PDF] Contents - OSU Math
. Proof. By Morera's theorem, it suffices to show that the integral of f around any triangular contour in D vanishes. If the triangle lies entirely inside ...
[16]
[PDF] Chapter 2 Complex Analysis
In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis.
[17]
[PDF] Complex Analysis in C 1.1 Holomorphic Functions - DPMMS
The uniform convergence on γ implies that R γ f dz = 0. Hence Morera's theorem implies that f is analytic on D. Cauchy's representation theorem shows that.
[18]
[PDF] Exercise on uniform convergence - Harold P. Boas
Sep 22, 2003 · If a sequence of holomorphic functions converges locally uniformly on an ... Morera's theorem (for instance), because a closed curve γ is a ...
[19]
[PDF] A rapid review of complex function theory 1 Holomorphic functions
(3) It suffices to prove uniform convergence on D(c;ρ) where ρ < R. On this ... Theorem 4.1 (Morera's theorem) If f is a continuous function on the ...
[20]
[PDF] Topologies on the Space of Holomorphic Functions (REVISED) - arXiv
Jul 23, 2007 · If a sequence of functions {fj} ⊆ O(Ω) converges in the topology T , then it converges uniformly on compact sets. The main tool in the proof of ...
[21]
[PDF] Ahlfors, Complex Analysis
... Morera's theorem, and it can be stated as follows: If f(z) is defined and ... RIEMANN ZETA FUNCTION. The series 1: n-u converges uniformly for all real ...
[22]
[PDF] John B. Conway, Functions of One Complex Variable, Springer ...
Morera's Theorem and Goursat's Theorem. Morera's Theorem. If G is a region and f : G → C a continuous function such that. R. T f dz = 0 for each triangular ...
[23]
Édouard Goursat (1858 - 1936) - Biography - MacTutor
Édouard Goursat attended the Collège de Brive-la-Gaillarde, receiving both his primary and secondary education there. He then spent one year at the Lycée Henri ...Missing: Morera 1888
[24]
Math 246A, Notes 3: Cauchy's theorem and its consequences
Oct 2, 2016 · Another nice consequence of the Cauchy integral formula is a converse to Cauchy's theorem known as Morera's theorem. Theorem 35 (Morera's ...