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Cauchy Integral TheoremEdit

The Cauchy Integral Theorem is one of the foundational results in Complex analysis. It concerns the behavior of contour integrals of holomorphic functions over closed curves in a region where the function is well-behaved. At its core, the theorem says that, under suitable conditions, the integral of a holomorphic function around a closed loop vanishes. This deceptively simple fact has far-reaching consequences, underpinning the way we understand analytic functions, contour integration, and several major techniques in the subject.

Historically, the theorem emerged from the work of mathematicians in the early 19th century as the calculus of complex variables began to solidify. It was given a full, rigorous formulation by Augustin-Louis Cauchy and later strengthened by proofs due to Cauchy–Goursat theorem and others, which removed some earlier heuristic assumptions. Today the Cauchy Integral Theorem is standard equipment in any treatment of holomorphic function and is closely tied to other central results such as the Cauchy Integral Formula and the Residue Theorem.

Cauchy Integral Theorem

Statement

Let U be an open subset of the complex plane, and let γ be a closed, piecewise smooth curve contained in U. If f is holomorphic on an open set that contains γ and its interior, then the contour integral of f around γ is zero: ∮_γ f(z) dz = 0. In particular, if U is simply connected (or equivalently if γ is null-homotopic in U), then the integral over any closed curve γ in U vanishes. This makes the integral path-independent for holomorphic functions in simply connected regions and is a direct consequence of the analytic structure of f.

Context and equivalent formulations

  • The Cauchy–Goursat formulation emphasizes that the conclusion holds without assuming any bound on the derivative of f; what is needed is merely holomorphy on a region containing the curve and its interior.
  • A common way to state the theorem is in relation to the existence of primitives: if the integral around every closed curve in a region vanishes, then f has a primitive (an antiderivative) on that region.
  • The result is a cornerstone behind the Cauchy Integral Formula, which expresses the values of a holomorphic function inside a region in terms of its values on the boundary of a contour surrounding the point of interest. See Cauchy Integral Formula for the explicit formula and its consequences.

Proof ideas (high level)

  • A standard route reduces the problem to a triangulation of the region and then applies the one-step case on small triangles, where holomorphy implies the integral around the boundary of each triangle is zero. Summing over the partition shows the integral around the outer boundary is zero.
  • Morera’s theorem, which provides a converse in a different direction, states that if the integral of f over every triangle in a region is zero, then f is holomorphic. See Morera's theorem for a related perspective.

Examples

  • If f is a polynomial, such as f(z) = z^n, then for any closed loop γ contained in a region where f is holomorphic, ∮_γ f(z) dz = 0. This illustrates the general statement in a concrete setting.
  • A function like f(z) = 1/z is holomorphic on C \ {0}, but it is not holomorphic on a region that contains a loop encircling the origin and its interior. Consequently, ∮_γ (1/z) dz around such a loop need not be zero (indeed, it equals 2πi for a unit circle around the origin). This highlights the necessity of the interior being contained in the domain of holomorphy.

Relations and consequences

  • The theorem underpins the Cauchy Integral Formula, which rewrites the values of f inside a contour in terms of the values on the contour.
  • It is a stepping stone toward the Residue Theorem and many other results in analytic function theory, including the maximum modulus principle and analytic continuation.
  • The idea of path-independence of contour integrals for holomorphic functions in simply connected regions leads to powerful tools for evaluating integrals and understanding the global behavior of analytic functions.

Generalizations and variants

  • For multiply connected domains, the integral of a holomorphic function around a closed curve need not vanish unless the curve is null-homologous in the domain. The theorem thus emphasizes the topological as well as analytic nature of holomorphic functions.
  • The core principle generalizes to different classes of contours (e.g., piecewise smooth, rectifiable) and to various regions in the complex plane where holomorphy holds.
  • Extensions and related results include Morera’s theorem discussed above, as well as the broader framework of analytic continuation and the interplay with the Cauchy–Riemann equations.

See also