Complex Analysis

Standard complex analysis topics from the canonical curriculum.

Core Topics

• Polar & Exponential Form — Modulus, argument, and Euler's formula z = r e^iθ.
• De Moivre & Roots of Unity — Powers and nth-roots of complex numbers via polar form.
• Cauchy-Riemann Equations — The partial-derivative test for when a complex function is holomorphic.
• Holomorphic & Analytic Functions — Complex differentiability, Cauchy-Riemann, and the analytic equivalence.
• Contour Integration — Parametrize the path, then integrate along it in the complex plane.
• Cauchy's Integral Theorem — A holomorphic function's loop integral is zero.
• Cauchy's Integral Formula — Recover a holomorphic function and its derivatives from boundary values.
• Laurent Series — Two-sided expansion on an annulus, and what the principal part says about a singularity.
• Singularities & Poles — Removable, pole, or essential — read it off the Laurent principal part.
• Residue Theorem — Contour integrals as 2π i times the sum of enclosed residues.
• Residues & Real Integrals — Compute hard real integrals by closing a contour and summing residues.
• Argument Principle & Rouché — Count zeros minus poles by contour integral; locate zeros with Rouché.
• Möbius Transformations — Fractional linear maps on the Riemann sphere and the cross-ratio.
• Liouville's Theorem — A bounded entire function is constant — via the Cauchy estimate.
• Maximum Modulus Principle — Nonconstant holomorphic |f| peaks only on the boundary.
• Harmonic Functions & Conjugates — Laplace's equation, harmonic parts, and building the conjugate.

Not sure where to start? Take the ten-question placement test.