A self-study journey in pure math

Complex Analysis

Compute the example, see how it's visualized, then prove it.

Start with Lesson 0 → How I built this

The background is live domain coloring, with each point z colored by arg f(z) for a rational function whose zero is drifting around the plane.

Built as I learned it

After learning the basics of high school calculus, I got fixated on complex functions. The idea that there was a whole field of math on numbers that aren't even Real fascinated me. After many hours of staring at complex-analysis.com, I knew I didn't have the tools to explore the subject yet, but I would definitely be back. Finishing Multivariable and learning enough vector calculus by the end of my senior year finally let me tackle this topic: what analysis on complex functions actually looks like.

I'm Max, an incoming math major at UCSB's College of Creative Studies. This site isn't a textbook; much better ones already exist. It's a record of the path I carved through the subject: the lessons I wanted to learn, as I actually learned them, bumps included.

The system behind it (session structure, spaced review, a running profile of how I learn, and the actual prompts that commissioned each visualization) is written up in How I built this.

Lessons

All lessons →

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

Behind the scenes

How I built this

The actual workflow: compute → visualize → prove, Socratic sessions, spaced review, and the real prompts that commissioned each visualization.

Read the writeup →

The gallery

Visualizations

Every lesson embeds its interactives where they belong, next to the math they illustrate. All 18 also live in one gallery if you just want to play.

Browse all 18 →

Just for fun

Dancing Geometric Series

The geometric series, dancing to music. Drop in any song and watch the partial sums of Σ zⁿ move to it, one spiral per frequency band, live-tunable.

Make it dance →