Lessons

0 ℂ Lesson 0 — The Big Picture: What Is Complex Analysis About? Where I started: a complex function maps a plane to a plane, so an honest graph would need four dimensions. Domain coloring and grid deformation are the workaround pictures — every later lesson leans on them. → 1 ℂ Lesson 1 — Complex Numbers, the Plane, and Basic Topology The ground floor. Arithmetic and geometry of complex numbers, polar form, roots of unity, and just enough topology — open sets, boundaries, regions — to state the later theorems honestly. → 2 ∂ Lesson 2 — Holomorphic Functions & the Cauchy-Riemann Equations The first real theorem. What complex differentiability actually demands, the Cauchy-Riemann equations that fall out of it, and why passing that test forces a map to preserve angles. → 3 Σ Lesson 3 — Power Series & Elementary Functions Power series take over: $e^z$ defined by its series, Euler's formula derived instead of assumed, and the logarithm turning multivalued — the first branch cut. → 4 ∮ Lesson 4 — Cauchy's Theorem & the Cauchy Integral Formula The centerpiece. Contour integrals, deformation invariance, Cauchy's theorem, and the integral formula: the values of a holomorphic function on a boundary determine every value inside. → 5 ∮ Lesson 5 — Consequences of Cauchy's Formula The payoff. One formula cascades into infinite differentiability, Cauchy's estimates, Liouville's theorem, and a proof of the Fundamental Theorem of Algebra with no algebra in it. → 5a ℝ Lesson 5a — The Real-Analysis Bridge The bridge. Five lessons in, the proofs had shifted from computing things to bounding things — so before series took over the subject, I paused to build the analysis underneath: sup and inf, rigorous limits, the triangle inequality, and the ML-estimate derived from scratch. → 6 Σ Lesson 6 — Power Series Representations & Analytic Continuation The biggest structural fact yet: a holomorphic function equals its own Taylor series. That forces zeros to be isolated, gives at most one analytic continuation, and turns a singularity into something with an exact, measurable location — removable, pole, or essential. → 7 Res Lesson 7 — Residues & Contour Integration Of the infinitely many coefficients in a Laurent series, exactly one is worth computing for contour integration. That single coefficient becomes the Residue Theorem, and — via a semicircular contour and an ML-estimate — a genuine way to evaluate a real integral with no antiderivative in sight. → 8 ↦ Lesson 8 — Conformal Mapping & Möbius Transformations What complex functions *do* to shapes. Möbius transformations send circles and lines to circles and lines, are conformal everywhere, and are pinned down by three points — building to the Riemann Mapping Theorem, where nearly every simply connected region turns out to be the unit disk in disguise. → 9 Γ Lesson 9 — The Gamma Function An explicit, computable extension of the factorial to almost all of the complex plane — built from one integration by parts, one Gaussian integral, and an algebraic chain that reaches every non-integer point in finitely many steps, ending in a reflection formula that ties Γ to sine itself. → 10 ζ Lesson 10 — The Riemann Zeta Function & the Riemann Hypothesis A series that converges nowhere useful, extended to the whole plane and made to say something about prime numbers. Basel's problem falls out of a residue computation, the Euler product links the series to the primes, and the whole apparatus builds to a precise statement of what the Riemann Hypothesis actually claims. → 11 ∼ Lesson 11 — Saddle-Point Method & Generating Functions Every previous lesson used a series to understand a function. This one runs the machinery backwards: a sequence encoded as a function, its nearest singularity read off as a growth rate, and Stirling's formula derived twice over, once along the real line and once through a saddle point in the complex plane. → 12 Res Lesson 12 — Branch Cuts & the Keyhole Contour Lesson 7's semicircular contour rested on three quiet assumptions. This one breaks single-valuedness: branch cuts built from the ground up, the keyhole contour that turns the branch jump into the answer, and the integral that Lesson 9's reflection formula imports from outside. →