Once you allow the single new number i with i^2 = -1, a whole continent of mathematics opens up. Complex analysis is the calculus of functions that eat a complex number and return one — f : \mathbb{C} \to \mathbb{C} — and it is one of the most elegant, surprising and useful theories there is. Demanding that such a function be differentiable turns out to be an extraordinarily strong condition, and from it flows a cascade of almost magical results.

This course assumes you already know complex numbers (the algebra of a + bi, the complex plane, and Euler's formula) and the real calculus of limits, derivatives, integrals and series. It then builds the complex theory on top, one small step at a time.

The big surprise: differentiable once ⟹ differentiable forever

In real calculus a function can be differentiable once but not twice, smooth but not analytic. In the complex world that never happens. A function that is complex-differentiable on an open region (we call it holomorphic) is automatically infinitely differentiable, equals its own Taylor series, and is pinned down completely by its values on any tiny contour. That single rigidity is the secret engine behind Cauchy's theorem, the residue calculus, and a tool so sharp it evaluates real integrals that defeat every method of ordinary calculus.

The shape of the journey

Five stages, from a single complex function to the geometry of conformal maps.

• Stage 1 — Functions of a complex variable. What it means to differentiate in \mathbb{C}, the Cauchy–Riemann equations, and holomorphic functions.
• Stage 2 — Complex integration. Integrating along contours, and Cauchy's theorem and integral formula — the heart of the subject.
• Stage 3 — Series & singularities. Power and Laurent series, the points where functions blow up, and the all-important residue.
• Stage 4 — The residue theorem. One theorem that turns hard real integrals into a sum of residues.
• Stage 5 — Conformal mappings. Holomorphic functions as angle-preserving maps, Möbius transformations, and physics on reshaped domains.

Stage 1 — Functions of a complex variable

A function w = f(z) maps the plane to the plane. When is it differentiable — and why is that so special?

1. Complex Functions
2. Limits and Continuity in ℂ
3. The Complex Derivative
4. The Cauchy–Riemann Equations
5. Holomorphic Functions
6. The Complex Exponential and Logarithm

Stage 2 — Complex integration

Integrate a complex function along a path, and discover that for holomorphic functions the path barely matters at all.

1. Contour Integration
2. Cauchy's Integral Theorem
3. The ML Inequality
4. Cauchy's Integral Formula
5. Analytic ⟹ Infinitely Differentiable

Stage 3 — Series & singularities

Every holomorphic function is a power series; near a singularity it is a Laurent series, whose key coefficient is the residue.

1. Complex Power Series
2. Taylor Series in ℂ
3. Laurent Series
4. Classifying Singularities
5. Residues

Stage 4 — The residue theorem

The pay-off: a contour integral is just 2\pi i times the sum of the residues inside — and that cracks real integrals wide open.

1. The Residue Theorem
2. Real Integrals by Residues
3. Trigonometric Integrals
4. The Argument Principle

Stage 5 — Conformal mappings

Holomorphic functions preserve angles. That geometric fact reshapes hard problems into easy ones.

1. Conformal Maps
2. Möbius Transformations
3. Applications of Conformal Maps

Let's get started

We begin where all of calculus begins — by asking what a function does, and what it means for it to have a derivative. Only now the input is a complex number, and the answer is far richer.

Let's get started → Complex Functions