Unit · year 2
MU-206 · Complex Analysis
Threads change · space26 lectures6 theorems
How complex differentiability forces astonishing rigidity and power.
Lectures
| L01 | Complex Numbers, Modulus, and Argument — |
| L02 | The Complex Plane and the Riemann Sphere — |
| L03 | Complex Functions and Continuity — |
| L04 | Complex Differentiability |
| L05 | The Cauchy–Riemann Equations |
| L06 | Harmonic Functions and Conjugates |
| L07 | Elementary Functions: exp, log, and Branch Cuts — |
| L08 | Conformal Mapping — |
| L09 | Möbius Transformations — |
| L10 | Contour Integration — |
| L11 | The ML Inequality and Estimates — |
| L12 | Cauchy's Integral Theorem |
| L13 | Deformation of Contours and Homotopy |
| L14 | Cauchy's Integral Formula |
| L15 | Derivatives of All Orders |
| L16 | Liouville's Theorem |
| L17 | The Fundamental Theorem of Algebra |
| L18 | Power Series and Analyticity — |
| L19 | Taylor and Laurent Expansions — |
| L20 | Classification of Singularities |
| L21 | Residues and Their Computation |
| L22 | The Residue Theorem |
| L23 | Evaluating Real Integrals by Residues |
| L24 | The Argument Principle and Rouché's Theorem — |
| L25 | Analytic Continuation — |
| L26 | Synthesis: The Rigidity of the Holomorphic |
Theorems in this unit
T-068
The Cauchy–Riemann equations
Complex differentiability is equivalent to a pair of PDEs.
T-069
Cauchy's integral theorem
The integral of a holomorphic function round a closed loop is zero.
T-070
Cauchy's integral formula
A holomorphic function's values are determined by its boundary values.
T-071
Liouville's theorem
A bounded entire function is constant.
T-072
The fundamental theorem of algebra
Every non-constant polynomial over C has a root.
T-073
The residue theorem
A contour integral equals the sum of enclosed residues.