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Cauchy's integral theorem

Cauchy's integral theorem, also known as the Cauchy–Goursat theorem, is a central result in stating that if a function f(z) is holomorphic in a simply connected open set U \subseteq \mathbb{C}, then for every simple closed contour \gamma lying in U, the contour integral \oint_\gamma f(z) \, dz = 0. This theorem establishes the path independence of integrals of holomorphic functions in simply connected domains and forms the cornerstone for many subsequent developments in the field. The theorem was first proved by the French mathematician in 1825 as part of his foundational work on complex integration, initially assuming the continuity of the function's derivative. In 1884, Édouard Goursat provided a refined proof that eliminated the need for continuity of the derivative, relying solely on the existence of the complex derivative, thus strengthening the result and making it applicable to a broader class of holomorphic functions. Cauchy's original contributions built on his earlier 1814 studies of definite integrals, which laid the groundwork for treating complex functions rigorously. Beyond its statement, the theorem has profound implications for , enabling the evaluation of without explicit antiderivatives and underpinning key results such as the Cauchy integral formula, which expresses the value of a at a point inside a in terms of its , and the for computing around singularities. It also connects to through concepts like and winding numbers, allowing to depend only on the topological class of the path. These properties make the theorem indispensable for applications in physics, , and , including solving differential equations and studying .

Introduction

Overview and importance

Cauchy's integral theorem is a fundamental result in that asserts: if a function f is throughout a simply connected \Omega and C is a simple closed curve in \Omega, then \int_C f(z) \, dz = 0. This theorem establishes that the of a holomorphic function over any closed path within such a domain vanishes, mirroring the behavior of conservative vector fields in real where path independence holds for exact differentials. The theorem's importance lies in its role as a cornerstone of complex analysis, enabling the development of powerful tools such as the residue theorem and Cauchy's integral formula, which simplify the evaluation of contour integrals that are often intractable in real analysis. By demonstrating that holomorphic functions possess inherently path-independent integrals in simply connected regions, it underpins much of the field's analytic machinery, including series expansions, conformal mappings, and applications in physics and engineering. In contrast to , where closed-path of differentiable functions do not necessarily vanish and depend on the specific path taken, Cauchy's theorem highlights the rigidity and global properties arising from complex differentiability. A key example illustrating the theorem's limitations is the function $1/z, which is holomorphic everywhere except at the origin; its around the unit circle |z|=1 equals $2\pi i, demonstrating that the result fails in domains that are not simply connected, such as the punctured plane.

Historical development

The foundations of Cauchy's theorem were laid by in his 1814 memoir on definite integrals, with further developments in the early as part of his pioneering efforts to establish rigorous . In his 1821 textbook Cours d'analyse de l'École Royale Polytechnique, Cauchy introduced precise definitions of limits, , and , providing the analytical framework essential for later developments in complex function theory. This work emphasized the importance of strict inequalities and , shifting toward modern rigor, though it focused primarily on real variables. Cauchy's specific contributions to complex integration emerged in his 1825 memoir Mémoire sur les intégrales définies prises entre des limites imaginaires, where he extended the of definite integrals to limits and stated the theorem that the integral of an over a closed vanishes, provided the function's is continuous. However, this original formulation and proof relied on the restrictive assumption of for the , which limited its generality and applicability to broader classes of holomorphic functions. In 1884, Édouard Goursat strengthened the theorem by providing a proof that eliminated the continuity assumption on the derivative, requiring only that the function be analytic inside and on a simple closed . Published as "Démonstration du théorème de Cauchy" in Acta Mathematica, this refinement, now known as the Cauchy-Goursat theorem, established the result in its modern form and became a cornerstone of . Cauchy's theorem profoundly influenced subsequent advancements in , including Bernhard Riemann's work on multi-valued functions and the development of Riemann surfaces in the mid-19th century.

Mathematical foundations

Prerequisite concepts

A , also known as an , is a that is complex differentiable at every point in its domain. Specifically, a function f is holomorphic at a point z_0 if the complex derivative f'(z_0) exists, defined as the f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}, where h is a approaching 0 from any direction in the . This definition requires the limit to be independent of the path taken by h, distinguishing complex differentiability from real differentiability, where directional limits may differ. A function is holomorphic on an if it is holomorphic at every point in that set. For a function f(z) = u(x, y) + i v(x, y), where z = x + i y and u, v are real-valued functions, the existence of the complex derivative implies the Cauchy-Riemann equations: \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. Conversely, if the partial derivatives exist at the point and the Cauchy-Riemann equations hold there, then f is complex differentiable at that point. Furthermore, if the partial derivatives exist and are continuous in an open neighborhood of the point and satisfy the Cauchy-Riemann equations there, then f is holomorphic on that neighborhood. Examples include polynomials like f(z) = z^2, which satisfy these conditions everywhere, and the f(z) = e^z, which is entire (holomorphic on the whole ). Contour integrals form another foundational tool in complex analysis, generalizing line integrals to the complex plane. For a function f(z) continuous on a contour C—a piecewise smooth curve in the complex plane parametrized by z(t) = x(t) + i y(t) for t \in [a, b]—the contour integral is defined as \int_C f(z) \, dz = \int_a^b f(z(t)) z'(t) \, dt, where z'(t) = x'(t) + i y'(t). In terms of real and imaginary parts, with f(z) = u(x, y) + i v(x, y) and dz = dx + i dy, this expands to \int_C f(z) \, dz = \int_a^b \left[ u(x(t), y(t)) + i v(x(t), y(t)) \right] \left( x'(t) + i y'(t) \right) dt. This parametrization allows evaluation of integrals along curved paths, such as circles or line segments, and is independent of the specific parametrization as long as it traces C in the same direction. Simply connected domains provide the topological setting for many results in . A (an open connected set) in the is simply connected if it has no holes, meaning every simple closed curve in the encloses only points within the and can be continuously contracted to a point without leaving the . For example, an open disk is simply connected, as any closed curve inside it can be shrunk to its center, whereas the punctured plane \mathbb{C} \setminus \{0\} is not, since a circle around the origin cannot be contracted to a point without crossing the puncture. This property ensures that closed contours behave predictably, distinguishing simply connected regions from multiply connected ones like annuli.

Statement of the theorem

Cauchy's integral theorem states that if f is holomorphic throughout a simply connected \Omega \subseteq \mathbb{C} and C is a positively oriented, piecewise smooth, simple closed in \Omega, then the of f over C is zero: \int_C f(z) \, dz = 0. This requires f to be holomorphic at every point inside the region bounded by C as well as on C itself. A closed contour C is a continuous in \mathbb{C} that is piecewise smooth and returns to its starting point, thus enclosing a bounded . The positive orientation refers to the counterclockwise traversal of C, which aligns with the for the interior . As an illustrative example, consider f(z) = z, which is holomorphic everywhere in \mathbb{C}, on the simply connected unit disk \Omega = \{ z \in \mathbb{C} : |z| < 1 \}. Let C be the boundary |z| = 1, parametrized by z(\theta) = e^{i\theta} for $0 \leq \theta \leq 2\pi, so dz = i e^{i\theta} \, d\theta. The integral becomes \int_C z \, dz = \int_0^{2\pi} e^{i\theta} \cdot i e^{i\theta} \, d\theta = i \int_0^{2\pi} e^{2i\theta} \, d\theta = i \left[ \frac{e^{2i\theta}}{2i} \right]_0^{2\pi} = 0, verifying the theorem.

Formulations and extensions

Simply connected domains

In complex analysis, a domain \Omega \subset \mathbb{C} is defined as simply connected if it is a non-empty, connected open set such that every closed curve in \Omega is contractible to a point within \Omega, meaning it can be continuously deformed to a single point without leaving the domain. This property ensures that the domain has no "holes" that prevent such deformations. For example, the open unit disk \{ z \in \mathbb{C} : |z| < 1 \} is simply connected, as any closed curve within it can be shrunk to the origin while remaining inside the disk. In contrast, the annulus \{ z \in \mathbb{C} : r < |z| < R \} for $0 < r < R is not simply connected, because a closed curve circling the inner boundary cannot be contracted to a point without crossing the excluded disk \{ |z| \leq r \}. Cauchy's theorem in the context of simply connected domains states that if f is holomorphic on a simply connected domain \Omega, then for any closed contour C in \Omega, \int_C f(z) \, dz = 0. This result holds because the simply connectedness allows the use of homotopy arguments to deform C to a point, making the integral vanish for holomorphic functions, which possess local antiderivatives. The condition is crucial, as it guarantees path independence of integrals and the existence of a global antiderivative in \Omega. Without simply connectedness, holomorphic functions may not satisfy this integral property over all closed curves. A classic counterexample illustrates the necessity of this condition: consider f(z) = 1/z, which is holomorphic on the punctured plane \mathbb{C} \setminus \{0\}, a domain that is not simply connected due to the hole at the origin. The integral over the unit circle |z| = 1, traversed counterclockwise, is \int_{|z|=1} \frac{1}{z} \, dz = 2\pi i \neq 0, demonstrating that the theorem fails in non-simply connected regions where curves can encircle singularities outside the domain. The role of homotopy here is evident: in simply connected domains, all closed curves are homotopic to a point (winding number zero around any exterior point), enabling the integral to be zero, whereas in domains like the punctured plane, non-trivial homotopy classes around the origin prevent this.

General topological conditions

The homotopy version of Cauchy's theorem generalizes the classical result to arbitrary domains by leveraging the topological notion of homotopy between curves. Specifically, if f is holomorphic in an open set \Omega \subseteq \mathbb{C} and \gamma_0, \gamma_1 are closed curves in \Omega that are homotopic in \Omega, then \int_{\gamma_0} f(z) \, dz = \int_{\gamma_1} f(z) \, dz. This equality holds because a homotopy provides a continuous deformation between the curves, allowing the integral to be deformed continuously without altering its value, provided f remains holomorphic throughout the region swept by the deformation. A key implication of this formulation is that the integral of a holomorphic function over any closed curve vanishes if that curve is homotopic to a constant curve (i.e., a point) within the domain. In such cases, the curve can be continuously shrunk to a point without leaving \Omega, rendering the integral zero by the homotopy invariance. This extends the theorem beyond simply connected domains, where every closed curve is homotopic to a point, to more general settings where only specific curves satisfy this contractibility condition. For multiply connected domains, which contain "holes" or singularities that prevent full contractibility, the theorem applies under additional restrictions: the integral over a closed curve C vanishes if C does not enclose any singularities of f. One common approach to handle such domains is to introduce cuts or slits connecting the singularities to the boundary, effectively rendering the domain simply connected for the purpose of integration along C, or to use principal branches of the function to avoid encircling isolated points. This ensures the curve remains in a region where homotopy to a point is possible relative to the singularities. To quantify these topological constraints more precisely, the winding number provides a fundamental tool. For a closed curve \gamma and a point a \notin \gamma, the winding number is defined as n(\gamma, a) = \frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z - a}, which measures how many times \gamma winds around a. The integral \int_{\gamma} f(z) \, dz = 0 for holomorphic f in the relevant domain if and only if n(\gamma, a) = 0 for all singularities a of f. This integer-valued invariant captures the topological obstruction in multiply connected settings, linking the theorem directly to the global structure of the domain.

Proofs

Proof via Green's theorem

The proof of Cauchy's theorem via Green's theorem relies on real analysis techniques and assumes that the complex function f is continuously differentiable (hence holomorphic) inside and on a simple closed positively oriented curve C, with the interior domain D bounded by C being simply connected. Decompose f(z) = u(x, y) + i v(x, y), where u and v are real-valued functions that are continuously differentiable in D. The complex line integral then separates into real and imaginary parts: \int_C f(z) \, dz = \int_C (u \, dx - v \, dy) + i \int_C (v \, dx + u \, dy). This follows from substituting dz = dx + i \, dy and expanding the product. Apply Green's theorem to each line integral over the region D: \int_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA, where the theorem holds under the continuity assumptions on the partial derivatives. For the real part, set P = u and Q = -v: \int_C u \, dx - v \, dy = \iint_D \left( -\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \, dA. For the imaginary part, set P = v and Q = u: \int_C v \, dx + u \, dy = \iint_D \left( \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) \, dA. Green's theorem applies directly due to the continuous differentiability of u and v. Since f is holomorphic, u and v satisfy the Cauchy-Riemann equations (as detailed in the prerequisite concepts section): \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. Substituting into the real part integrand yields -\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = - \left( -\frac{\partial u}{\partial y} \right) - \frac{\partial u}{\partial y} = \frac{\partial u}{\partial y} - \frac{\partial u}{\partial y} = 0. For the imaginary part, \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} = \frac{\partial v}{\partial y} - \frac{\partial v}{\partial y} = 0. Thus, both double integrals vanish, implying \int_C f(z) \, dz = 0.

Goursat's proof without continuity

In 1884, Édouard Goursat published a proof of Cauchy's integral theorem that eliminates the need for continuity of the derivative f', assuming only its existence (holomorphicity) in a simply connected domain. This resolved a subtlety in Cauchy's original 1825 argument, where continuity of f' was implicitly invoked but not fully justified. The proof relies on two core elements: Goursat's lemma, which establishes the theorem for triangular contours, and a triangulation of the domain to extend the result to general closed contours. Goursat's lemma states that if f is holomorphic in an open set containing a closed triangle \Delta and its interior, then \int_{\partial \Delta} f(z) \, dz = 0. To prove this, proceed by contradiction. Suppose \left| \int_{\partial \Delta} f(z) \, dz \right| = \varepsilon > 0, where \Delta has side length h. Connect the midpoints of \Delta's sides to subdivide it into four smaller triangles \Delta_1, \Delta_2, \Delta_3, \Delta_4, each of side length h/2. The integrals over the internal segments cancel in pairs, so \int_{\partial \Delta} f(z) \, dz = \sum_{k=1}^4 \int_{\partial \Delta_k} f(z) \, dz. Thus, \left| \sum_{k=1}^4 I_k \right| = \varepsilon, where I_k = \int_{\partial \Delta_k} f(z) \, dz, implying \max_k |I_k| \geq \varepsilon / 4 (since \varepsilon \leq \sum |I_k| \leq 4 \max |I_k|). Select the \Delta_1 achieving this maximum and repeat the subdivision, yielding a sequence of nested triangles \Delta_n with side lengths h_n = h / 2^n \to 0 and integrals I_n satisfying |I_n| \geq \varepsilon / 4^n. The triangles \Delta_n converge to a point z_* \in \Delta where f is differentiable. Near z_*, f(z) = f(z_*) + f'(z_*) (z - z_*) + \varepsilon_n(z) (z - z_*), with \varepsilon_n(z) \to 0 uniformly on \Delta_n as n \to \infty. Then, \int_{\partial \Delta_n} f(z) \, dz = f(z_*) \int_{\partial \Delta_n} dz + f'(z_*) \int_{\partial \Delta_n} (z - z_*) \, dz + \int_{\partial \Delta_n} \varepsilon_n(z) (z - z_*) \, dz. The first two integrals vanish over the closed contour, leaving |I_n| \leq \sup_{\Delta_n} |\varepsilon_n| \cdot \sup_{\Delta_n} |z - z_*| \cdot \length(\partial \Delta_n) \leq 3 \delta_n h_n^2, where \delta_n = \sup |\varepsilon_n| \to 0. Since h_n = h / 2^n, this upper bound is $3 \delta_n h^2 / 4^n. For sufficiently large n, $3 \delta_n h^2 < \varepsilon, so |I_n| < \varepsilon / 4^n, contradicting |I_n| \geq \varepsilon / 4^n. Thus, \varepsilon = 0, proving the lemma. To extend to a general simple closed contour C enclosing a simply connected region D where f is holomorphic, triangulate the interior of C: divide D into finitely many triangles \Delta_j (possible since D is polygonalizable and triangulable) such that the union of their boundaries consists of C plus internal segments traversed oppositely in pairs. Then, \int_C f(z) \, dz = \sum_j \int_{\partial \Delta_j} f(z) \, dz, as internal integrals cancel. By Goursat's lemma, each \int_{\partial \Delta_j} f(z) \, dz = 0, so the total integral vanishes. No mesh refinement is needed, as the lemma applies exactly to each finite triangle.

Applications and consequences

Derivation of Cauchy's integral formula

Let f be holomorphic in an open domain containing a positively oriented simple closed contour C and its interior, with a a point in the interior of C. Consider the auxiliary function g(z) = \frac{f(z) - f(a)}{z - a} for z \neq a. Since f is differentiable at a, \lim_{z \to a} g(z) = f'(a), so the apparent singularity of g at a is removable, and g extends holomorphically to the closed disk bounded by C. By Cauchy's theorem, as the interior of C is simply connected and g is holomorphic there, \int_C g(z) \, dz = 0. Substituting the expression for g yields \int_C \frac{f(z) - f(a)}{z - a} \, dz = 0, which rearranges to \int_C \frac{f(z)}{z - a} \, dz = f(a) \int_C \frac{dz}{z - a}. The integral \int_C \frac{dz}{z - a} = 2\pi i. This holds because \frac{1}{z - a} is holomorphic in the region between C and a small circle \gamma_\epsilon of radius \epsilon > 0 around a; by Cauchy's theorem applied to this annular region, the integrals over C and -\gamma_\epsilon are equal. Parametrizing \gamma_\epsilon as z = a + \epsilon e^{i\theta}, dz = i \epsilon e^{i\theta} \, d\theta for $0 \leq \theta \leq 2\pi, gives \int_{\gamma_\epsilon} \frac{dz}{z - a} = \int_0^{2\pi} i \, d\theta = 2\pi i, independent of \epsilon. Thus, \int_C \frac{f(z)}{z - a} \, dz = 2\pi i f(a), or equivalently, Cauchy's integral formula f(a) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - a} \, dz. The formula extends to higher derivatives by differentiating with respect to a under the sign, justified by the of the derivatives on compact subsets of the of holomorphy. Differentiating once yields f'(a) = \frac{1}{2\pi i} \int_C \frac{f(z)}{(z - a)^2} \, dz. By , the n-th satisfies f^{(n)}(a) = \frac{n!}{2\pi i} \int_C \frac{f(z)}{(z - a)^{n+1}} \, dz, or \frac{f^{(n)}(a)}{n!} = \frac{1}{2\pi i} \int_C \frac{f(z)}{(z - a)^{n+1}} \, dz. For the entire function f(z) = e^z, the formula applied at a = 0 with C the unit circle gives \frac{f^{(n)}(0)}{n!} = \frac{1}{2\pi i} \int_C \frac{e^z}{z^{n+1}} \, dz, which equals the coefficient of z^n in the of e^z. This recovers $1/n!, aligning with the series e^z = \sum_{n=0}^\infty \frac{z^n}{n!}, and demonstrates how the integral formula generates power series expansions for holomorphic functions.

Connection to the residue theorem

The residue theorem extends Cauchy's theorem to handle functions with isolated singularities inside a closed contour, providing a way to evaluate contour integrals by summing contributions from those singularities. Specifically, if f(z) is analytic inside and on a simple closed positively oriented contour C, except for finitely many isolated singularities a_k inside C, then \oint_C f(z) \, dz = 2\pi i \sum_k \operatorname{Res}(f, a_k), where \operatorname{Res}(f, a_k) denotes the residue of f at a_k. This result builds directly on Cauchy's theorem, which asserts that the integral vanishes when there are no singularities inside C in a simply connected domain; in such cases, the residues at any potential points are zero, recovering the zero integral. The residue at an a is defined as the a_{-1} in the of f around a, f(z) = \sum_{n=-\infty}^\infty a_n (z - a)^n, with \operatorname{Res}(f, a) = a_{-1}. This captures the "principal part" of the singularity's contribution to the integral, as the integral over a small contour around a equals $2\pi i \, a_{-1}, and the full residue theorem sums these for all interior singularities. For example, consider f(z) = \frac{1}{z(z-1)} and the contour C: |z| = 2, which encloses the simple poles at z = 0 and z = 1. The residue at z = 0 is \lim_{z \to 0} z \cdot \frac{1}{z(z-1)} = \frac{1}{-1} = -1, and at z = 1 it is \lim_{z \to 1} (z-1) \cdot \frac{1}{z(z-1)} = \frac{1}{1} = 1. Thus, the sum of residues is -1 + 1 = 0, and \oint_C f(z) \, dz = 2\pi i \cdot 0 = 0. This illustrates how the residue theorem accounts for the cancellation of contributions from multiple singularities, yielding a zero integral in this case, in contrast to Cauchy's theorem applied in singularity-free regions.