What you'll learn

44 lessons in Complex analysis

Complex functionsHolomorphic functionsContour integrals & residuesPower series & Taylor expansionSingularities classifiedConformal mappingsArgument principle & winding numbersCauchy's integral formulaLiouville & fundamental theorem of algebraMaximum modulus principleAnalytic continuationHarmonic functionsRiemann surfaces (intro)Riemann mapping theoremPicard's great theoremProof of Cauchy-Riemann equationsProof: Liouville → FTAProof of Cauchy integral formulaRiemann zeta functionBranch cuts & multivalued functionsRouché's theoremSchwarz lemma & automorphismsMontel's theorem & normal familiesMittag-Leffler & Weierstrass factorizationThe Gamma functionEvaluating real integrals by residuesThe hypergeometric function ₂F₁Legendre polynomials & functionsBessel functionsWeierstrass elliptic function ℘Jacobi theta functionsAsymptotic expansions & steepest descentSchwarz–Christoffel mappingThe Phragmén–Lindelöf principleOrder & Hadamard factorizationModular forms & the modular groupThe Mellin transformDirichlet problem & the Poisson kernelComplex dynamics: Julia & Mandelbrot setsRunge's approximation theoremUnivalent functions & the Bieberbach conjectureSplit-complex numbersTwo routes to π/eA sum whose length changes under the integral sign

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Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Complex analysis exactly where it's useful.

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Related Analysis subjects

Calculus Multivariable calculus Differential equations Partial Differential Equations Dynamical Systems & Chaos Real analysis