Mathematics / advanced
Complex functions and analytic structure
Use holomorphicity to connect local series, contour integration, global geometry, and transform methods.
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Module 01 advanced
Complex functions and analytic structure
Use holomorphicity to connect local series, contour integration, global geometry, and transform methods.
- 01
Complex differentiability and the Cauchy–Riemann equations
Recognize a complex derivative as one real-linear approximation compatible with every approach direction.
27 min - 02
Contour integrals and Cauchy's theorem
Compute oriented integrals and distinguish local holomorphicity from the global conditions that make closed integrals vanish.
28 min - 03
Cauchy's integral formula and derivative estimates
Recover interior derivatives from surrounding values and use contour bounds to prove analytic rigidity.
29 min - 04
Analytic continuation and identity theorems
Use interior accumulation and overlapping analytic descriptions without confusing one Taylor disk with an entire function's domain.
28 min - 05
Laurent series and isolated singularities
Classify punctured-neighborhood behavior using negative powers and keep each expansion tied to its annulus.
29 min - 06
Residue calculus and real integrals
Reduce suitable contour integrals to local coefficients and justify the limiting arcs needed for real integrals.
30 min - 07
The argument principle and Rouché's theorem
Count zeros with multiplicity using logarithmic derivatives and boundary inequalities rather than solving for every root.
29 min - 08
Conformal maps and the Schwarz lemma
Separate local angle preservation from global one-to-one behavior and derive the rigidity of normalized disk maps.
29 min - 09
Harmonic functions and Dirichlet problems
Connect holomorphic real parts to Laplace's equation and solve the continuous boundary-value problem on a disk.
30 min - 10
Branches, monodromy, and Riemann surfaces
Track path-dependent continuation, distinguish a chosen cut from a singularity, and make multivalued expressions single-valued on a suitable surface.
31 min