Cauchy Integral FormulaEdit

The Cauchy Integral Formula is one of the core results in complex analysis. It reveals a striking rigidity of holomorphic functions: if a function is analytic on and inside a simple closed curve, then its values inside the curve are completely determined by its values along the boundary. Named after Augustin-Louis Cauchy, the formula sits at the heart of many powerful techniques, including contour integration, Taylor series representations, and derivative estimates, and it plays a central role in both theory and applications.

In its simplest and most widely used form, the formula asserts that for a function f holomorphic on an open set containing a positively oriented simple closed contour γ and its interior, and for any z inside γ, the value of f at z is given by a boundary integral: f(z) = (1 / (2π i)) ∮γ f(w) / (w − z) dw. From this single identity, a cascade of important consequences follows, such as ways to compute derivatives inside the region and to bound those derivatives in terms of boundary data. The method extends beyond a single derivative to higher derivatives, yielding a family of formulas that tie f^{(n)}(z) to integrals over γ: f^(n)(z) = n! / (2π i) ∮γ f(w) / (w − z)^(n+1) dw for n ≥ 0.

Statement

Let γ be a positively oriented, simple closed curve in an open set Ω ⊆ C, and suppose f is holomorphic on Ω and on and inside γ. If z is a point in the interior of γ, then f(z) = (1 / (2π i)) ∮γ f(w) / (w − z) dw. Moreover, for any integer n ≥ 0 and the same hypotheses, the higher-order formula holds: f^(n)(z) = n! / (2π i) ∮γ f(w) / (w − z)^(n+1) dw.

The integrals are taken with respect to w along γ in the positive (counterclockwise) orientation. The contour γ need not be a circle; it can be any simple closed curve as long as it encloses z and lies within the domain of holomorphy of f.

Notation and hypotheses

f must be holomorphic on an open set containing γ and its interior.
γ is oriented positively; z is in the interior of γ.
The integrals use the variable w to emphasize that the integration is around the boundary with respect to the boundary parameter.

Remarks

The formula shows that holomorphic functions are real-analytic in the strongest possible sense: they are determined by their boundary values and by contour data.
The same principle underlies many techniques in contour integration and is closely related to the residue theory that formalizes the contribution of a simple pole at w = z.

Geometric and analytic interpretation

The Cauchy Integral Formula encodes a boundary-to-interior principle. Because f is holomorphic, its behavior inside the region is not independent of boundary values; rather, the entire interior can be reconstructed from the boundary data via the integral. This mirrors the rigid structure of holomorphic functions: they are infinitely differentiable and equal to their Taylor series expansions within their domain of holomorphy.

In particular, the formula provides a direct route to Taylor series representations. If f is analytic near z0, one can choose a small contour γ encircling z0 and, by expanding f(w) in powers of (w − z0) and exchanging summation and integration, deduce that the local power series coefficients are given by derivatives of f at z0: f(z) = Σ_{n=0}^∞ f^(n)(z0) / n! · (z − z0)^n.

Linkages to other core ideas include Cauchy integral theorem (a tool used in establishing CIF) and holomorphic function theory, where the boundary-to-interior principle is a recurring theme.

Proof sketch

A standard route to the Cauchy Integral Formula uses the residue theorem or, equivalently, a direct application of Cauchy’s theorem to the integrand f(w)/(w − z). Since f is holomorphic on and inside γ, the only singularity of the integrand within γ is the simple pole at w = z, with residue f(z). By the residue theorem, the contour integral around γ picks up 2π i times that residue, yielding the stated equality: ∮_γ f(w) / (w − z) dw = 2π i · f(z). Dividing by 2π i gives the formula. The higher-derivative version follows by applying the same idea to f(w)/(w − z)^(n+1) and noting the corresponding residue at w = z.

Alternatively, one can derive CIF by considering the Cauchy integral of f and using Cauchy's differentiation under the integral sign, justified by uniform convergence on compact subsets of Ω.

Generalizations and related results

Higher derivatives: The n-th derivative version above generalizes CIF to obtain f^(n)(z) directly from boundary data.
Taylor and Laurent theory: CIF provides a constructive route to Taylor series in a neighborhood of any interior point and connects to the broader framework of analytic expansion.
Estimates for derivatives (Cauchy estimates): From CIF one derives bounds such as |f^(n)(z)| ≤ n! M / r^n, where M is a bound on |f| on a circle of radius r around z, which in turn underpins convergence and growth properties.
Extensions to more general domains: While CIF is stated for simply connected regions with a simple closed contour, analogous ideas appear in more general settings (for example, using homology of contours and multiple connected domains), and they underpin many results in complex analysis.
Connections to contour integration and residue theory: CIF is part of the same toolkit as the Residue theorem and the broader theory of Contour integration.

Examples and applications

Simple check: If f(w) = 1, then f(z) = (1 / (2π i)) ∮_γ 1/(w − z) dw = 1, as expected.
Exponential function: For f(w) = e^w and z inside γ, CIF gives f(z) = (1 / (2π i)) ∮_γ e^w / (w − z) dw, recovering e^z from boundary data.
Derivatives via CIF: The first derivative formula f'(z) = (1 / (2π i)) ∮_γ f(w) / (w − z)^2 dw shows how differentiation can be transferred to the boundary integral, a principle that underlies many computational techniques.
Computation of real integrals: CIF is a central tool in the method of contour integration used to evaluate certain real integrals by closing a contour in the complex plane and summing residues.

History

The Cauchy Integral Formula is part of the historical development of complex analysis in the 19th century, with foundational work by Augustin-Louis Cauchy and a broader consolidation by later mathematicians who clarified the analytic structure of holomorphic functions and the interplay between boundary data and interior values.