---
title: "Complex Analysis"
description: "12 topics in Mathematics, in the order they build on each other."
canonical: https://lightmysky.com/learn/mathematics/areas/complex-analysis
source: https://lightmysky.com/learn/mathematics/areas/complex-analysis.md
retrieved: 2026-09-02
---

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# Complex Analysis

12 topics in Mathematics, in the order they build on each other.

Page: https://lightmysky.com/learn/mathematics/areas/complex-analysis

- [Complex Functions and the Complex Plane as a Domain](https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs): Treat a function of a complex variable as a map of the plane to itself, and set up limits and continuity in that setting.
- [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss): Demand a derivative independent of the direction of approach, and derive the two partial differential equations that demand forces.
- [Harmonic Functions and What Analyticity Forces](https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4): Show the real and imaginary parts of an analytic function each satisfy Laplace's equation, and recover one from the other.
- [Contour Integrals Along Parametrised Paths](https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P): Integrate a complex function along a path by parametrising it, and bound the result by length times maximum modulus.
- [Cauchy's Theorem and Deforming a Contour](https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH): Prove that an analytic function integrates to zero around a closed loop it is analytic inside, and use that to slide contours freely.
- [Cauchy's Integral Formula and Derivatives of Every Order](https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ): Recover a function's value inside a contour from its values on the contour, then differentiate under the integral to get every derivative at once.
- [Power Series and Analyticity in the Complex Plane](https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8): Expand an analytic function as a power series on a disc, and read the radius of convergence off the nearest singularity.
- [Laurent Series and Classifying Isolated Singularities](https://lightmysky.com/learn/mathematics/laurent-series-and-classifying-isolated-singularities-mt_WoQV0gD1O-): Allow negative powers to expand a function on an annulus, and sort singularities by how the negative part behaves.
- [The Residue Theorem](https://lightmysky.com/learn/mathematics/the-residue-theorem-mt_BN2biZ_zos): Reduce an integral around a closed contour to a sum of residues, one number per singularity enclosed.
- [Evaluating Real Integrals by Residues](https://lightmysky.com/learn/mathematics/evaluating-real-integrals-by-residues-mt_Lj_HXtjzAh): Close a real integral into a contour in the plane, discard the added arc with an estimate, and read the answer off the residues.
- [The Argument Principle and Rouche's Theorem](https://lightmysky.com/learn/mathematics/the-argument-principle-and-rouches-theorem-mt_J1dC428M84): Count zeros and poles inside a contour by watching how the argument turns, and transfer a count to a nearby function.
- [Conformal Maps and Mobius Transformations](https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI): See that an analytic map with nonzero derivative preserves angles, and use Mobius maps to move standard regions onto one another.