Complex analysis is the study of functions of complex variables, extending calculus to the complex plane where differentiation and integration acquire fundamentally richer structure than their real counterparts. A function that is complex-differentiable — called holomorphic or analytic — satisfies a stringent set of conditions encoded in the Cauchy-Riemann equations, and this rigidity forces remarkable consequences: infinite differentiability, exact integral evaluation via residues, and the ability to conformally reshape domains. The field underpins signal processing (Laplace and Z-transforms), fluid mechanics, electrostatics, quantum field theory, and analytic number theory. Mastering complex analysis unlocks elegant proofs — such as deriving the sum \sum 1/n^2 = \pi^2/6 via residues — and provides geometric intuition for conformal mappings that transform one domain into another while preserving angles.
What This Cheat Sheet Covers
This topic spans 14 focused tables and 89 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Complex Number Fundamentals
Every complex number has an algebraic rectangular form and a geometric polar form; fluency with both is essential before tackling any deeper topic in complex analysis.
| Concept | Example | Description |
|---|---|---|
z = a + bi,\; i^2=-1 | • Standard form • a=\operatorname{Re}(z), b=\operatorname{Im}(z)• every complex number is a point or vector in \mathbb{C} | |
\lvert z\rvert = \sqrt{a^2+b^2} | • Distance from origin • \lvert z_1 z_2\rvert = \lvert z_1\rvert\lvert z_2\rvert and \lvert z_1+z_2\rvert \le \lvert z_1\rvert+\lvert z_2\rvert (triangle inequality) | |
\arg(z)=\theta = \operatorname{atan2}(b,a) | • Angle from positive real axis • multi-valued — principal value \operatorname{Arg}(z)\in(-\pi,\pi] | |
\bar{z} = a - bi | • Reflection across real axis • z\bar{z}=\lvert z\rvert^2• \operatorname{Re}(z)=\tfrac{z+\bar{z}}{2}, \operatorname{Im}(z)=\tfrac{z-\bar{z}}{2i} |