SYSTEMATIC MATHEMATICS

Complex Analysis

This course contains 38 visible knowledge chapters in nine content-sized units. A complex number is first treated as a point in a plane, and every new symbol is translated into distance, angle, direction, or local change before it enters a formula. The course asks what changes when a function must have the same derivative from every direction, then builds the answer through Wirtinger derivatives, contour integrals, Cauchy theory, power and Laurent series, zeros, singular values, maps, and real-integral contours. The Riemann sphere, winding number, root monodromy, Morera’s converse, Taylor radius, Casorati–Weierstrass density, the residue at infinity, open mapping and local inverses, Schwarz lemma, and Jordan decay close gaps that a short contour-trick course leaves hidden. Every theorem names its domain and hypotheses; every integral records path, orientation, poles, branches, and estimates; every formula is followed by a hand-checkable example and a failure boundary.

Before this course: Completed Calculus and Real Analysis. The course assumes real and multivariable derivatives, sequences and series, uniform convergence, compactness and connectedness in the real line and plane, Riemann integration, improper integrals, Taylor remainder reasoning, and proof with quantified limits. Complex arithmetic and polar form are reconstructed before use. No topology course, measure theory, differential equations, numerical software, physics transform course, or prior contour integration is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Cartesian and polar forms encode algebra and geometry

Objective: Why can the theorem in “Cartesian and polar forms encode algebra and geometry” not be reused blindly in this nearby case: Argument is multivalued globally; writing arg z as one continuous function around the origin requires a branch cut. Name the first failed hypothesis.

A complex number z=x+iy is a point/vector in R² with multiplication that adds angles and multiplies magnitudes.

Objects and notation: For z=x+iy, conjugate z̄=x−iy, modulus |z|=√(x²+y²), and nonzero z=re^{iθ} with θ defined modulo 2π. The result to establish is: |zw|=|z||w| and arguments add modulo 2π; z^{-1}=z̄/|z|² for z≠0. Read every formula on its stated domain and branch, and retain contour orientation and singularities.

The worked question is: Write −1+i√3 in polar form with principal angle. Start by “Modulus is √(1+3)=2.” and finish by “Thus z=2e^{2πi/3}; all arguments are 2π/3+2πk.” Then compare the answer with this nearby failure: Argument is multivalued globally; writing arg z as one continuous function around the origin requires a branch cut.

|zw|=|z||w| and arguments add modulo 2π; z^{-1}=z̄/|z|² for z≠0.

Expand zw and compute its squared modulus.

Factor (x²+y²)(u²+v²).

Use Euler form to identify angle addition and inverse.

Write −1+i√3 in polar form with principal angle.

  1. Modulus is √(1+3)=2.
  2. The point lies in quadrant II with angle 2π/3.
  3. Thus z=2e^{2πi/3}; all arguments are 2π/3+2πk.

Result: The principal polar form is 2e^{2πi/3}.