Complex Analysis

Why one extra dimension makes analysis rigid, and powerful

About this course

Holomorphic functions, Cauchy's theorem and integral formula, Laurent series and residues, and conformal mapping.

What you will be able to do

  • State the Cauchy-Riemann equations and explain why holomorphy is restrictive
  • Apply Cauchy's theorem and integral formula, and their consequences
  • Expand functions in power and Laurent series and classify singularities
  • Evaluate real integrals by residues
  • Use the argument principle and construct conformal maps

What this course covers

  • The Argument Principle: statement and proof
  • Why the result is an integer: the argument principle name
  • Stability and winding-number interpretation
  • Rouché's theorem and the fundamental theorem of algebra
  • The open mapping theorem and maximum modulus principle
  • Working procedure for counting zeros in a region
  • Closing synthesis: continuity plus integrality equals constancy
  • Every analytic function equals its Taylor series
  • Cauchy tradition vs. Weierstrass tradition reconciled
  • Radius of convergence and how to find it
  • Behaviour on the circle of convergence
  • Analytic continuation
  • Working procedure for computing radii
  • Worked example: radius for a rational function
  • The Residue Theorem: Statement and Proof
  • Computing Residues Fast: Three Techniques
  • Three Traps in Residue Computation
  • Pole Order of a Quotient
  • Residue at Infinity and the Zero-Sum Theorem
  • Working Procedure for Contour Integrals
  • Contour Integration as Finite Algebraic Reduction
  • Historical motivation: the cubic formula
  • Construction of ℂ as pairs of reals
  • ℂ cannot be ordered: loss of ordering
  • Multiplication as rotation and scaling (polar form)
  • Fundamental Theorem of Algebra and algebraic closure
  • ℂ as a metric space; failure of differentiability
  • Worked example: solving the cubic via ℂ
  • The central result: integral of 1/z equals 2πi
  • Definition of the contour integral
  • Path dependence and primitives
  • The ML inequality (estimation lemma)
  • Powers of z and the uniqueness of the 2πi anomaly
  • Working procedure and capstone example
  • Cauchy's Theorem: statement and context
  • Simply connected regions and the role of topology
  • Goursat's theorem: bisection proof on a triangle
  • From triangle to region: building a primitive on a convex set
  • Contour deformation
  • Cauchy's integral formula (preview)
  • Working procedure and summary
  • Why a complex function has no drawable graph
  • f(z) = z² as a map of the plane
  • Three substitutes for the graph
  • The complex exponential and its grid action
  • The complex logarithm and branch cuts
  • Sine is unbounded in the complex plane
  • 1/z sends lines to circles; the Riemann sphere
  • Contour integration strategy overview
  • Big-semicircle method for rational functions
  • Unit-circle substitution for trigonometric integrals
  • Principal value versus improper integral
  • Jordan's lemma for oscillatory integrands
  • Indented contour for a pole on the path
  • Worked example and master procedure summary
  • Möbius Transformations as Matrix Groups
  • Standard Dictionary of Conformal Maps
  • Conformal vs. Bijective Maps (Local vs. Global)
  • Riemann Mapping Theorem
  • Harmonicity Preserved Under Conformal Maps (Dirichlet Problem)
  • Accounting: Real-Analysis Surprises Explained by Complex Analysis
  • Master Statement: One Hypothesis, Nine Consequences
  • Laurent Series and the Annulus of Convergence
  • Historical Note: Laurent, Weierstrass, and Publication
  • Classification of Isolated Singularities
  • Behavioral Theorems for Each Singularity Type
  • Picard's Great Theorem and Essential Singularities
  • The Residue: The One Surviving Coefficient
  • Worked Classification and Residue Computation
  • Complex differentiability: the every-direction demand
  • Conjugation fails complex differentiability everywhere
  • |z|² is differentiable at exactly one point
  • The Cauchy–Riemann equations
  • Cauchy–Riemann necessary but not sufficient: the G counterexample
  • Harmonic functions and Laplace's equation
  • Worked procedure and rigidity consequences
  • Cauchy's Integral Formula: Statement and Proof
  • Mean Value Property and Maximum Modulus Principle
  • Historical Context: Cauchy vs. Weierstrass Traditions
  • Differentiation Under the Integral Sign and Infinite Differentiability
  • Applying the Formula to Evaluate Contour Integrals
  • Cauchy's Estimates
  • Liouville's Theorem and the Fundamental Theorem of Algebra
  • Conformality: angles preserved by analytic maps
  • Local magnification and area scaling
  • Failure of conformality at critical points
  • Conformal equivalence with analytic maps (converse and Möbius maps)
  • Orthogonal grid of level curves
  • Harmonic functions preserved under analytic composition
  • Worked example and master summary
  • Cauchy estimate: the unifying tool
  • Entire functions: definition and examples
  • Liouville's theorem and polynomial-growth generalisation
  • Fundamental theorem of algebra
  • Morera's theorem
  • Uniform limits of analytic functions are analytic
  • The identity theorem
  • Worked example and practical summary

Syllabus

  1. Holomorphic Functions

    1. L1 · Why One Dimension Was Not Enough
    2. L2 · Functions You Cannot Graph
    3. L3 · Cauchy-Riemann, and Why It Is So Restrictive
    4. L4 · Conformality, and the First Consequences

  2. Cauchy's Theorem

    1. L5 · Integrating Along a Curve
    2. L6 · Cauchy's Theorem
    3. L7 · Cauchy's Integral Formula
    4. L8 · What Follows Immediately

  3. Series and Singularities

    1. L9 · Power Series, and Why the Radius Is What It Is
    2. L10 · Laurent Series and the Classification of Singularities

  4. Residues and Their Uses

    1. L11 · The Residue Theorem
    2. L12 · Real Integrals, Done in the Plane

  5. Zeros and Conformal Mapping

    1. L13 · Counting Zeros: the Argument Principle and Rouche
    2. L14 · Conformal Mapping, and What It All Explained