# Cauchy's Theorem and Integral Formula *The pair of results that turn holomorphy into a global constraint: closed integrals of holomorphic functions vanish, and the value of a holomorphic function in a region is fixed by its values on the boundary.* ## Cauchy's theorem > **Cauchy's theorem (Cauchy–Goursat).** Let f be holomorphic on a > simply connected open set D ⊆ ℂ. Then for every closed piecewise- > smooth contour γ ⊂ D, > > ``` > ∮_γ f(z) dz = 0 > ``` "Simply connected" means D has no holes — every closed loop in D can be continuously shrunk to a point without leaving D. Shooting a pole into the middle of a disc destroys simple connectedness and breaks the theorem (witness ∮ dz/z = 2πi, Chapter 5). ### Sketch of proof via Green's theorem For f = u + iv with u, v ∈ C¹, Green's theorem in the plane gives, for any simple closed curve γ bounding a region R ⊂ D, ``` ∮_γ (u dx − v dy) = −∬_R (∂v/∂x + ∂u/∂y) dA ∮_γ (v dx + u dy) = ∬_R (∂u/∂x − ∂v/∂y) dA ``` Both integrands vanish identically on R by the Cauchy–Riemann equations (Chapter 4). Therefore ``` ∮_γ f dz = ∮_γ (u dx − v dy) + i ∮_γ (v dx + u dy) = 0 ``` Goursat's contribution was to prove this without assuming continuity of the partial derivatives — a technical strengthening needed to establish the holomorphy-implies-analyticity theorem of Chapter 3. ### Path-independence corollary If f is holomorphic on a simply connected D, and γ₁, γ₂ are two paths in D with the same endpoints, then γ₁ − γ₂ is a closed loop and ``` ∫_{γ₁} f dz − ∫_{γ₂} f dz = ∮_{γ₁ − γ₂} f dz = 0 ``` so line integrals depend only on the endpoints. The holomorphic function therefore admits a **primitive** F with F′ = f, defined by integration from a fixed base point. ### Deformation of contours In a multiply connected region — for example the annulus containing a single pole of f — Cauchy's theorem still gives a useful conclusion. Two closed contours γ₁, γ₂ both enclosing the pole with the same orientation are homotopic through the punctured region, so ``` ∮_{γ₁} f dz = ∮_{γ₂} f dz ``` This is why the unit-circle calculation of Chapter 5 returns 2πi for any loop enclosing z = 0: they are all deformations of each other. ## The Cauchy integral formula The crown jewel of the subject, and the workhorse of K–K. > **Cauchy integral formula.** Let f be holomorphic on an open set > containing a simple closed contour γ and its interior, with γ > positively oriented. Then for any point z₀ in the interior, > > ``` > f(z₀) = (1 / 2πi) ∮_γ f(z) / (z − z₀) dz > ``` The integrand has a simple pole at z₀ lying inside γ; the formula says the enclosed pole's contribution **reconstructs f at that point** from boundary data alone. The value of a holomorphic function anywhere inside a region is determined by its values on the boundary. ### Sketch of proof Shrink γ to a small circle C_ε of radius ε centred at z₀ (Cauchy's theorem plus contour deformation). Parametrise C_ε as `z = z₀ + ε e^{iθ}`, dz = `iε e^{iθ} dθ`: ``` (1 / 2πi) ∮_{C_ε} f(z)/(z − z₀) dz = (1/2π) ∫_0^{2π} f(z₀ + ε e^{iθ}) dθ ``` As ε → 0, the integrand → f(z₀) by continuity, so the limit equals f(z₀). This same limit argument appears again, doubled in half, in the indentation step of the K–K derivation (Chapter 8). ### The Cauchy formula for derivatives Differentiating the integral formula n times under the integral (which is legal because the kernel is smooth in z₀ when z₀ is in the interior) yields ``` f^{(n)}(z₀) = (n! / 2πi) ∮_γ f(z) / (z − z₀)^{n+1} dz ``` so **every** derivative of f at an interior point is determined by boundary values. This is the formula that shows holomorphic ⇒ infinitely differentiable ⇒ analytic (Chapter 3). ## Mean-value property, maximum modulus, Liouville Three immediate consequences worth naming. **Mean-value property.** Specialising the integral formula to a circle of radius r around z₀: ``` f(z₀) = (1 / 2π) ∫_0^{2π} f(z₀ + r e^{iθ}) dθ ``` The value at the centre equals the average over the boundary. Applied to u = Re f, this is the classical mean-value property of harmonic functions. **Maximum-modulus principle.** If f is holomorphic on a connected open set D and |f| attains a local maximum at an interior point of D, then f is constant on D. Non-trivial holomorphic functions have no interior local maxima of modulus. **Liouville's theorem.** Every bounded entire function is constant. The proof uses the derivative formula on a circle of radius R and lets R → ∞: |f′(z)| ≤ M/R → 0, so f′ ≡ 0. One-line corollary: the fundamental theorem of algebra — every non-constant polynomial has a complex root. ## How this is used in the K–K derivation Chapters 11–12 derive K–K by the following template, which is worth stating now so subsequent chapters feel motivated. 1. Let χ(ω) be the complex frequency response of a causal, linear, time-invariant system, analytic in the upper half-plane with χ → 0 at |ω| → ∞. 2. For a real frequency ω, the function `χ(ω′)/(ω′ − ω)` has a simple pole on the real axis at ω′ = ω and is otherwise analytic in the upper half-plane. 3. Integrate `χ(ω′)/(ω′ − ω)` around the closed contour Γ_R consisting of - the real axis from −R to ω − ε, - a small semicircle of radius ε in the UHP around ω (indentation), - the real axis from ω + ε to +R, - the large semicircle of radius R in the UHP. 4. By Cauchy's theorem the contour integral is **zero** (no poles enclosed, since we indented around the only real-axis pole). 5. The large semicircle → 0 as R → ∞ by Jordan's lemma (Chapter 9). 6. The small semicircle → −iπ χ(ω) as ε → 0 (half a residue, Chapter 7, Sokhotski–Plemelj, Chapter 8). 7. What remains is the principal-value integral along the real axis plus `−iπ χ(ω) = 0`, i.e. ``` P ∫_{−∞}^{∞} χ(ω′)/(ω′ − ω) dω′ = iπ χ(ω) ``` Separating real and imaginary parts yields the two K–K relations. Everything in that template is a straight-line consequence of Cauchy's theorem plus the ML / Jordan machinery. The Cauchy integral formula is the single result doing the heavy lifting. ## See also - [03_complex_functions_analyticity.md](03_complex_functions_analyticity.md) - [05_contour_integration.md](05_contour_integration.md) - [07_residues.md](07_residues.md) - [08_principal_value.md](08_principal_value.md)