Cauchy integral formula Study Guide

📖 Core Concepts Holomorphic ⇒ Analytic – A function that’s complex‑differentiable on a disk can be written as a convergent power series on that disk. Cauchy’s Integral Formula (CIF) – The value of a holomorphic function inside a curve is completely determined by its values on the boundary: \[ f(z{0})=\frac{1}{2\pi i}\oint{\gamma}\frac{f(z)}{z-z{0}}\,dz . \] Cauchy’s Differentiation Formula – All derivatives are obtained by the same contour integral with higher‑order poles: \[ f^{(n)}(z{0})=\frac{n!}{2\pi i}\oint{\gamma}\frac{f(z)}{(z-z{0})^{n+1}}\,dz . \] Uniform Limits Preserve Holomorphy – If a sequence of holomorphic functions converges uniformly on compact sets, the limit is holomorphic (follows from CIF). Cauchy’s Estimate – Bounds derivatives in terms of the maximum of \(|f|\) on a surrounding circle: \[ |f^{(n)}(a)|\le \frac{n!\,M}{R^{\,n}}\quad\text{if }|f(z)|\le M\text{ on }|z-a|=R . \] --- 📌 Must Remember CIF hypothesis: \(f\) holomorphic on the interior of \(\gamma\) and continuous on its closure; \(\gamma\) must have winding number 1 about \(z{0}\). Derivative formula coefficient: factor \(n!\) appears in the numerator. Cauchy’s Estimate ⇒ Liouville: Bounded entire \(\Rightarrow\) all derivatives zero \(\Rightarrow\) function constant. Gauss Mean‑Value Theorem: \(f(a)=\frac{1}{2\pi}\int{0}^{2\pi} f(a+Re^{i\theta})\,d\theta\). Infinite differentiability: Every holomorphic function is \(C^\infty\) (all derivatives exist via CIF). --- 🔄 Key Processes Applying CIF to compute \(f(z{0})\): Verify \(f\) holomorphic on region containing \(\gamma\) and interior. Ensure \(\gamma\) winds once around \(z{0}\). Evaluate the contour integral (often by parametrizing a circle). Deriving derivatives: Use differentiation formula with \(n=1,2,\dots\). For each \(n\), insert \((z-z{0})^{-(n+1)}\) into the integrand. Obtaining Cauchy’s estimate: Start from differentiation formula. Take absolute values and apply \(|\int|\le\) length × max |integrand|. Simplify using \(|z-z{0}|=R\) on the circle. Proving Liouville: Apply Cauchy’s estimate with \(R\to\infty\) for a bounded entire function. Conclude all derivatives vanish; function equals its constant value. --- 🔍 Key Comparisons CIF vs. Cauchy’s Differentiation Formula CIF gives \(f(z{0})\); differentiation formula gives \(f^{(n)}(z{0})\). Power of denominator: \(1/(z-z{0})\) vs. \(1/(z-z{0})^{n+1}\). Extra factor \(n!\) appears only in differentiation version. Holomorphic vs. Analytic Holomorphic: complex‑differentiable in an open set. Analytic: representable by a convergent power series. The theorem shows the two notions coincide on disks. --- ⚠️ Common Misunderstandings “Any closed curve works.” – The curve must have winding number 1 around the point; a curve that does not enclose \(z{0}\) gives zero integral. “Continuity of \(f\) on the interior is enough.” – Holomorphy on the interior is required; continuity alone does not guarantee CIF. “CIF only holds for circles.” – It holds for any rectifiable closed curve with the correct winding number. “Cauchy’s estimate gives equality.” – It provides an upper bound, not the exact derivative value. --- 🧠 Mental Models / Intuition “Boundary determines the interior.” – Think of a holomorphic function like a perfectly elastic membrane: knowing the edge (boundary values) fixes the whole shape. Residue as “charge” – The integral \(\frac{1}{2\pi i}\oint \frac{f(z)}{(z-z{0})^{n+1}}dz\) measures the “n‑th charge” (derivative) sitting at \(z{0}\). Cauchy’s estimate as “shrinking circle” – The farther the circle (larger \(R\)), the smaller the bound on higher derivatives, reflecting the smoothing effect of analyticity. --- 🚩 Exceptions & Edge Cases Non‑simple curves – If \(\gamma\) loops multiple times, the integral picks up the winding number \(w(\gamma,z{0})\): \[ f(z{0})=\frac{1}{2\pi i}w(\gamma,z{0})\oint{\gamma}\frac{f(z)}{z-z{0}}dz . \] Functions holomorphic only on punctured domains – CIF does not apply at isolated singularities; instead residues are used. --- 📍 When to Use Which Need the value of \(f\) at an interior point? → Use the basic CIF. Need a specific derivative \(f^{(n)}(z{0})\)? → Apply the differentiation formula. Estimating derivative size from a bound on \(|f|\)? → Use Cauchy’s estimate. Proving a bounded entire function is constant? → Combine Cauchy’s estimate with Liouville’s argument. Showing a limit of holomorphic functions is holomorphic? → Use uniform‑limit preservation via CIF. --- 👀 Patterns to Recognize Denominator power \((z-z{0})^{n+1}\) ↔ \(n^{\text{th}}\) derivative. Factor \(n!\) appears only when differentiating under the integral. Maximum‑modulus on a circle → derivative bound (Cauchy’s estimate pattern). Average over a circle equals center value – spot Gauss mean‑value pattern in problems asking for “average value of a holomorphic function”. --- 🗂️ Exam Traps Choosing the wrong contour: Selecting a curve that does not enclose \(z{0}\) or has winding number 0 yields zero, leading to an incorrect answer. Missing the \(n!\) factor: In derivative formulas, forgetting the factorial gives a value too small by that factor. Confusing boundedness with continuity: Liouville’s theorem requires the function to be entire (holomorphic everywhere), not merely continuous and bounded on a region. Applying Cauchy’s estimate with the wrong radius: Using the radius of a smaller circle than the one where the bound \(M\) holds weakens the estimate. Assuming any holomorphic function is automatically constant: Only when it is bounded on the whole complex plane (Liouville). ---