Complex analysis Study Guide

📖 Core Concepts Complex function \(f(z)=u(x,y)+i\,v(x,y)\): maps a complex number \(z=x+iy\) to another complex number; \(u,v\) are real‑valued functions of \((x,y)\). Holomorphic (analytic) function: has a complex derivative \(f'(z)\) at every point of an open set; automatically infinitely differentiable and equal to its Taylor series there. Cauchy–Riemann (CR) equations: necessary (and, with continuity of partials, sufficient) conditions for holomorphy: \[ ux = vy,\qquad uy = -vx . \] Meromorphic function: holomorphic except at isolated poles (points where the function blows up like \(\frac{1}{(z-a)^m}\)). Conformal map: a holomorphic function with non‑zero derivative; locally preserves angles and orientation. Residue at a pole \(z0\): coefficient \(a{-1}\) in the Laurent expansion; central to the Residue Theorem. --- 📌 Must Remember Differentiability ⇒ Analyticity for complex functions (true only in the complex setting). Liouville’s Theorem: a bounded entire function is constant. Cauchy Integral Theorem: \(\displaystyle \oint{\gamma} f(z)\,dz = 0\) for any closed contour \(\gamma\) lying in a holomorphic region. Cauchy Integral Formula: \[ f(z)=\frac{1}{2\pi i}\oint{\gamma}\frac{f(\zeta)}{\zeta-z}\,d\zeta . \] Residue Theorem: \[ \oint{\gamma} f(z)\,dz = 2\pi i\sum{k}\operatorname{Res}\bigl(f,zk\bigr). \] Laurent series near an isolated singularity \(z0\): \[ f(z)=\sum{n=-\infty}^{\infty}an (z-z0)^n \qquad(a{-1}=\operatorname{Res}(f,z0)). \] Picard’s Theorem (entire case): an entire non‑constant function attains every complex value, possibly omitting at most one point. Riemann Mapping Theorem: any non‑empty simply connected open set \(\neq\mathbb C\) can be conformally mapped onto the unit disk. --- 🔄 Key Processes Checking Holomorphy (CR test) Compute \(ux, uy, vx, vy\). Verify \(ux=vy\) and \(uy=-vx\). Ensure the partials are continuous (gives sufficiency). Finding a Residue at a Simple Pole \(z=a\) Write \(f(z)=\frac{g(z)}{(z-a)}\) with \(g\) holomorphic at \(a\). Residue = \(g(a)\). Residue at a Pole of Order \(m\) \[ \operatorname{Res}(f,a)=\frac{1}{(m-1)!}\,\lim{z\to a}\frac{d^{\,m-1}}{dz^{\,m-1}}\!\Big[(z-a)^m f(z)\Big]. \] Using the Cauchy Integral Formula (evaluating \(f^{(n)}(z0)\)) \[ f^{(n)}(z0)=\frac{n!}{2\pi i}\oint{\gamma}\frac{f(\zeta)}{(\zeta-z0)^{n+1}}\,d\zeta . \] Constructing a Laurent Series Identify the annulus of convergence around the singularity. Expand regular part as a Taylor series; expand singular part using geometric series or known expansions. Analytic Continuation Sketch Show two holomorphic functions agree on an overlapping open set → they agree on the whole connected domain (uniqueness). --- 🔍 Key Comparisons Holomorphic vs. Meromorphic Holomorphic: analytic everywhere in the domain. Meromorphic: analytic except at isolated poles. Real‑differentiable vs. Complex‑differentiable Real differentiable ⇒ existence of partials; does not guarantee complex derivative. Complex differentiable ⇒ CR equations hold → automatically analytic. Entire vs. Bounded Entire Entire: holomorphic on all of \(\mathbb C\). Bounded entire: by Liouville, must be constant. Simple Pole vs. Essential Singularity Simple pole: Laurent series has only a single \(\frac{a{-1}}{z-a}\) term. Essential: infinitely many negative‑power terms; behavior wildly oscillatory (Picard). Conformal Map vs. General Holomorphic Map Conformal: derivative never zero → angle‑preserving & orientation‑preserving. General holomorphic: may have zero derivative (critical points) → not conformal there. --- ⚠️ Common Misunderstandings “Partial derivatives exist ⇒ \(f\) is holomorphic.” Wrong: CR equations must also hold (and partials must be continuous). “Every analytic function is bounded.” Wrong: Only bounded entire functions are forced to be constant (Liouville). “Residue = value of the function at the pole.” Wrong: Residue is the coefficient of \((z-a)^{-1}\) in the Laurent expansion, not the function value (which is infinite). “Any simply connected region can be mapped to any other by a holomorphic function.” Wrong: Need a conformal bijection; the Riemann Mapping Theorem guarantees a map onto the unit disk, not arbitrary targets. “Orientation is irrelevant for conformal maps.” Wrong: Holomorphic maps preserve orientation; antiholomorphic maps reverse it. --- 🧠 Mental Models / Intuition Holomorphic = “smooth as a marble”: think of a complex function as a perfectly smooth surface that you can zoom in on forever, always seeing a perfect linear (rotation + scaling) approximation. Residue = “charge” at a pole: just as a point charge creates a flux, a pole contributes a fixed \(2\pi i\) amount to any contour integral encircling it. Cauchy Integral Formula = “averaging over a circle”: the value inside a disk is the weighted average of boundary values; the kernel \(\frac{1}{\zeta-z}\) acts like a “magnetic pull” toward the interior point. --- 🚩 Exceptions & Edge Cases Picard’s Theorem: an entire function can omit at most one complex value (e.g., \(e^{z}\) omits \(0\)). Poles vs. Removable Singularities: if the Laurent principal part is zero, the singularity is removable (extend the function). Connected vs. Simply Connected: Cauchy’s theorems require the domain to be simply connected for the integral to be zero; a domain with a hole can host non‑zero integrals even for holomorphic functions. Zero derivative points: a holomorphic function with \(f'(z0)=0\) is still holomorphic but not conformal at \(z0\). --- 📍 When to Use Which Cauchy Integral Formula – evaluate \(f(z0)\) or its derivatives when the function is holomorphic on and inside a simple closed contour. Residue Theorem – compute real integrals or contour integrals that encircle isolated poles; especially when the integrand is rational or has simple trigonometric forms. Laurent Series – analyze behavior near singularities, classify poles vs. essential singularities, or compute residues quickly. Conformal Mapping – transform a complicated boundary‑value problem (e.g., fluid flow around an airfoil) to a simpler geometry (unit disk). Analytic Continuation – extend a function defined by a power series or integral representation beyond its original radius of convergence (e.g., Riemann zeta). --- 👀 Patterns to Recognize Integrand of the form \(\frac{P(z)}{Q(z)}\) with \(\deg Q = \deg P + 2\) → integral over a large circle often vanishes (Jordan’s lemma). Simple pole at \(z=a\) ⇒ residue = \(\displaystyle\lim{z\to a}(z-a)f(z)\). Repeated factor \((z-a)^m\) in denominator → use derivative formula for residues (order‑\(m\) pole). Series \(\sum{n=0}^{\infty} an (z-z0)^n\) converges inside radius \(R\); if singularity appears at \(|z-z0|=R\), expect a Laurent expansion with negative powers just beyond. When a contour integral evaluates to \(2\pi i\) times something, look for a pole inside the contour. --- 🗂️ Exam Traps | Trap | Why It Looks Plausible | Correct Reasoning | |------|-----------------------|-------------------| | Assuming real differentiability ⇒ holomorphic | Partial derivatives exist ⇒ derivative exists (real intuition). | Need CR equations + continuity; many counter‑examples (e.g., \(f(z)=\overline{z}\)). | | Choosing Cauchy Integral Theorem for a domain with a hole | The theorem states “integral of holomorphic function around a closed curve is zero.” | The region must be simply connected; a hole can produce a non‑zero integral (e.g., \(\frac{1}{z}\) around the origin). | | Treating a removable singularity as a pole | The Laurent series has a term that looks like \(\frac{a{-1}}{z-a}\). | If the principal part is actually zero after simplification, the singularity is removable – no residue. | | Using Liouville without checking boundedness | “Entire ⇒ constant” is a common mis‑recall. | Liouville requires bounded entire; unbounded entire functions (e.g., \(e^{z}\)) are not constant. | | Confusing orientation in residue theorem | Many forget the sign; assume always \(+2\pi i\). | Positive (counter‑clockwise) orientation gives \(+2\pi i\); clockwise gives \(-2\pi i\). | | Believing every holomorphic function is conformal | “Holomorphic = angle‑preserving.” | Conformal requires \(f'(z)\neq0\); zeros of the derivative break conformality (critical points). | | Misidentifying essential singularities | “If series has negative powers, it must be a pole.” | Essential singularities have infinitely many negative‑power terms; poles have finitely many. | | Assuming analytic continuation always possible | “Every function can be continued.” | Continuation is unique only when overlapping region is non‑empty and the function is analytic there; sometimes branch cuts prevent global continuation. | ---