Chapter 02
Contour Integration
Complex line integrals, Cauchy's theorem for simply connected domains, the Cauchy integral formula, and Morera's converse.
Built · Not yet releasedCore analysis · Advanced undergraduate and graduate · Staged public deployment
Complex Analysis
Differentiable once, and therefore analytic forever.
Complex analysis is where analysis becomes geometry. A single complex derivative forces a function to be infinitely differentiable, rigid, and beautifully structured. This course develops holomorphic functions and the Cauchy-Riemann equations, contour integration and Cauchy's theorems, Taylor and Laurent series, the residue calculus and its power over real integrals, conformal mappings and the Riemann mapping theorem, and the global theory of entire and meromorphic functions.
6Chapters
24Visual investigations
GraduateLevel
Chapter 1 is free now.
All six chapters are built. Chapters 2 through 6 will be released progressively as public deployment continues.
Where this course sits
Real Analysis→General Topology→Complex Analysis→Functional Analysis→Riemann Surfaces
What you will master
Three capabilities the course builds
I
See functions as mappings
Read holomorphic functions geometrically as angle-preserving conformal maps of the plane, not merely as formulas.
II
Integrate by residues
Use Cauchy's theorem and the residue theorem to evaluate definite integrals that resist every real-variable method.
III
Command the rigidity of holomorphy
Exploit the extraordinary rigidity of holomorphic functions, from Liouville and the maximum principle to analytic continuation and Picard.
The syllabus
Six chapters, built in sequence
All six chapters are built. Chapter 1 is released free now. Chapters 2 through 6 remain visible here as the course roadmap and will be released progressively. Each chapter includes geometric visual investigations, complete definitions and proofs, guided exercises, progressive hints, and detailed corrections.
Chapter 01
Holomorphic Functions
Complex differentiability and the Cauchy-Riemann equations, harmonic functions and their conjugates, and the first examples: polynomials, the exponential, the trigonometric functions, and the complex logarithm.
Free · Open chapterChapter 03
Taylor and Laurent Series
Power series in a disk, the Taylor expansion of holomorphic functions, Laurent series in an annulus, and the classification of isolated singularities.
Built · Not yet releasedChapter 04
Residue Theorem and Applications
Residues at isolated singularities, the residue theorem as the culmination of Cauchy's theory, and the evaluation of real definite integrals that resist real-variable methods.
Built · Not yet releasedChapter 05
Conformal Mappings
Angle-preserving maps, Mobius transformations and their group structure, the Joukowski map, and the Riemann mapping theorem.
Built · Not yet releasedChapter 06
Entire and Meromorphic Functions
Liouville's theorem and the fundamental theorem of algebra, Weierstrass factorization, Hadamard's theorem, and the great and little theorems of Picard.
Built · Not yet releasedVisual first
Four geometric investigations per chapter, on light crisp backgrounds.
Complete proofs
Every theorem stated with all hypotheses and proved in full.
Guided practice
Exercises with progressive hints and a full worked correction.
Bilingual
Full English and French presentation sharing one mathematical corpus.