Complex Analysis
Why are complex functions so much better behaved?
A real function can be differentiable once and never again. A complex function differentiable once is differentiable infinitely often, determined by its values on any small disc, and impossible to bound without being constant. This subject is about why one extra dimension causes that.
Holomorphic functions
What does one extra dimension cost?
Differentiability, but from every direction at once
The complex derivative requires the same limit from infinitely many directions. That single demand is the source of everything strange in the subject.
Angles survive; distances do not
Holomorphic maps preserve angles and distort everything else, which is visible long before it is provable. The film is mostly animation for exactly this reason.
Contour integration
Why does the path not matter?
Integrals that forget the route
Two different paths between the same points give the same integral, which is false for real line integrals in general. Cauchy's theorem says when and why.
Residues
Can a hole be measured by walking around it?
What a loop knows about what it encircles
A closed path picks up information about singularities it never touches. The residue theorem turns that into a computation, and then into real integrals that look impossible.
Analytic continuation
How far can a function be extended before it contradicts itself?
Complex Analysis, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume VI ↓