Complex Analysis

Number Systems: From Naturals to Complex Numbers

A comprehensive journey through number systems from natural numbers to complex numbers. Definitions, properties, operations, complex conjugates, geometric interpretations, and algebraic structures. Read more... 2876 words,🕔14 minutes read, May 17, 2022.

Polar form & Euler's Formula

Exploring polar and exponential representations of complex numbers. Euler's formula, De Moivre's Theorem, roots of unity, and geometric interpretations of complex multiplication and division. Read more... 2438 words,🕔12 minutes read, May 17, 2022.

n-th Roots of Unity: Geometry, Properties, and Group Structure

A complete guide to n-th roots of unity. Derivation via Euler’s formula, geometric interpretation as a regular n-gon on the unit circle, primitive roots, algebraic properties, and connection to cyclic groups ℤ/nℤ. Read more... 2349 words,🕔12 minutes read, May 17, 2022.

Cyclotomic Polynomials & General N-th Roots of Complex Numbers

Cyclotomic polynomials and their factorization of xⁿ-1. General formula for extracting n-th roots of any complex number using roots of unity and Euler's formula Read more... 1420 words,🕔7 minutes read, May 17, 2022.

Complex-Valued Functions: Domain, Range, Classification, and Operations

Foundational article on complex-valued functions covering definitions, domain and range, algebraic vs transcendental classification, single vs multi-valued functions, entire functions and Liouville's theorem, and operations on complex functions. Read more... 3161 words,🕔15 minutes read, Jun 21, 2025.

Transcendental Functions

In-depth exploration of transcendental functions in complex analysis, covering the complex exponential, multi-valued logarithm and its principal branch, trigonometric and hyperbolic functions, power functions, and their fundamental properties including periodicity, branch cuts, mapping behavior, and unboundedness in the complex plane. Read more... 2300 words,🕔11 minutes read, Jun 21, 2025.

Circles, Polar Forms, and Roots in the Complex Plane

Circle equations |z − a| = r, interior/exterior regions, products and quotients in polar form, De Moivre's theorem, and n-th roots of complex numbers. Read more... 1573 words,🕔8 minutes read, Jun 17, 2025.

Stereographic Projection and the Riemann Sphere

Comprehensive derivation of stereographic projection formulas mapping the complex plane to the Riemann sphere, arithmetic operations with infinity, and the chordal metric on the extended complex plane. Read more... 1821 words,🕔9 minutes read, Jun 17, 2025.

Complex Square Root, Exponentiation, and the Quadratic Formula

In-depth exploration of complex square root (polar and Cartesian forms), complex exponentiation using the multi-valued logarithm, properties and caveats of these operations, and solving quadratic equations in the complex plane with discriminant analysis. Read more... 2359 words,🕔12 minutes read, Jun 23, 2025.

Basic Topology of the Complex Plane

A rigorous introduction to the topological foundations of complex analysis. Metrics and distance in ℂ, open/closed discs, neighborhoods (including punctured), boundary/interior/exterior/limit points, half-planes, annuli, sectors, strips, and the fundamental theorems characterizing open and closed sets. Read more... 2748 words,🕔13 minutes read, Jun 28, 2025.

Visualizing Complex Functions

Comprehensive guide to visualizing complex functions as geometric transformations, analyzing how translations, rotations, dilations, power functions, exponential and logarithmic mappings transform points, lines, circles, and regions between domain and codomain planes. Read more... 2259 words,🕔11 minutes read, Jun 23, 2025.

Closure, Compactness, and Connectedness in the Complex Plane

A comprehensive exploration of closure of sets, bounded and compact sets (including Heine-Borel theorem), connected and path-connected sets, domains and regions. Read more... 3178 words,🕔15 minutes read, Jun 28, 2025.

Interior Points and Open Sets

A rigorous exploration of interior points, open sets, and foundational topology in the complex plane. Includes definitions, characterizations, examples (disks, annuli, half-planes), counterexamples (rational points, boundaries), and proofs of key propositions. Read more... 2218 words,🕔11 minutes read, Jun 25, 2025.

Open Sets in C. Unions, Intersections, Continuity, and Representation

Rigorous treatment of open sets in the complex plane. Definitions, characterizations (interior equality, boundary exclusion, complement closure), and core topological properties. Proofs that arbitrary unions and finite intersections of open sets are open. Covers the continuity-preimage theorem, representation of open sets as unions of open disks, and comparative properties of open/closed sets. Read more... 1381 words,🕔7 minutes read, Jun 25, 2025.

Exterior, Limit, and Isolated Points

Definitions and fundamental relationships between exterior, limit, and isolated points in the complex plane. Includes analysis of open sets, punctured neighborhoods, and boundary classification. Read more... 2886 words,🕔14 minutes read, Jun 25, 2025.

Closure and Limit Points

Proof that the closure of any set in the complex plane is closed. Establishes the equivalence between a set being closed, containing all its limit points, and being equal to its closure. Includes detailed examples of limit points, such as the unit circle, isolated points, density of rationals in ℂ, and a set with exactly one limit point. Read more... 1884 words,🕔9 minutes read, Jun 28, 2025.

Bounded Sets, Compactness, and Connectedness in the Complex Plane

Comprehensive notes on topology in the complex plane, covering bounded sets, compactness via the Heine-Borel theorem, and connectedness through polygonal paths. Examines the hierarchy of domains, regions, star-shaped, and simply connected sets, including detailed proofs on finite sets and the annulus, plus applications in Cauchy’s Theorem and the Riemann Mapping Theorem. Read more... 3045 words,🕔15 minutes read, Jun 28, 2025.

Boundedness, Compactness & Connectivity in the Complex Plane

A comprehensive overview of key topological concepts in ℂ, bounded and compact sets (Heine–Borel theorem), the Bolzano–Weierstrass property, Cantor’s intersection theorem, compactness of closed subsets, definitions of connected regions and polygonal connectivity, and sequential convergence criteria (limits, Cauchy sequences). Read more... 4487 words,🕔22 minutes read, May 17, 2022.