Complex Analysis

Complex Functions

1. Complex Arithmetic & Geometrycomplex plane, polar form, roots of unity, Moivre theorem6.5–11.5 h
2. Complex Functions & Mappingspolynomials, exponential, log, trig, branches6.5–11.5 h
3. Analytic Functionsderivative in C, Cauchy-Riemann equations, harmonic functions6.5–11.5 h

Integration Theory

1. Contour Integralsparametrization, estimates, ML inequality6.5–11.5 h
2. Cauchy's Theoremhomotopy version, simply connected domains6.5–11.5 h
3. Cauchy Integral Formuladerivatives, Liouville, fundamental theorem of algebra, Morera6.5–11.5 h

Series & Singularities

1. Power Series & AnalyticityTaylor series, uniqueness of representation6.5–11.5 h
2. Laurent Series & Singularitiespoles, essential singularities, classification6.5–11.5 h
3. Residue Theoremresidue computation, evaluation of real integrals, summation of series6.5–11.5 h

Applications

1. Conformal MappingMobius transformations, Schwarz-Christoffel, solving boundary problems6.5–11.5 h
2. Argument Principle & Rouche's Theoremcounting zeros, stability criteria6.5–11.5 h
3. Applications to Physics & Engineeringpotential flow, electrostatics in 2D, signal processing6.5–11.5 h