Ready to begin. All 15 chapters are available — start with Chapter 1 below, or continue wherever you left off.
Numbers as locations on a map: modulus, argument, polar form, De Moivre's theorem, and the n-th roots.
Limits, continuity, and the strict meaning of differentiability when a function maps a plane to a plane.
The two-rule test for analyticity, derived from scratch. Point versus region, and harmonic functions for free.
Reverse the machinery: given one half of an analytic function, reconstruct the other. The Milne-Thomson method.
What analytic functions do to whole regions — stretching and rotating while preserving every angle.
The most famous family of maps in mathematics: circles to circles, three points to three points.
Integration along a path — and why, for analytic functions, the path stops mattering.
A function's interior fixed by its boundary — and the legendary theorems that fall out of it.
Represent, classify, extract, apply — power series to residues. The densest, most-tested topic in the course.
Slicing 2D regions and summing pillars. Setting up limits, swapping the order, and going polar.
Rotate your coordinates to match the symmetry. The Jacobian, and the sign errors that cost marks.
Volumes of awkward shapes. Three coordinate systems, three volume elements — and the spherical sin φ.
Walking through a force field, summing the push. Green's theorem turns a boundary walk into an area integral.
Flux through a curtain. Parametrising surfaces, computing dS, and linking a boundary curve to its surface.
The third great bridge: flux through a closed surface equals what the volume inside produces.