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A concise course in complex analysis and Riemann surfaces

A concise course in complexanalysis and Riemann surfacesWilhelm SchlagContentsPrefacevChapter 1. Fromitoz: the basics of complex analysis11. The field of complex numbers12. Differentiability and conformality33. M obius transforms74. Integration125. Harmonic functions196. The winding number217. Problems24 Chapter 2. Fromzto the Riemann mapping theorem: some finer points of basiccomplex analysis271. The winding number version of Cauchy s theorem272. Isolated singularities and residues293. Analytic continuation334. Convergence and normal families365. The Mittag-Leffler and Weierstrass theorems376. The Riemann mapping theorem417. Runge s theorem448. Problems46 Chapter 3. Harmonic functions onD511. The Poisson kernel512. Hardy classes of harmonic functions533. Almost everywhere convergence to the boundary data554.

Chapter 1. From ito z: the basics of complex analysis 1 1. The field of complex numbers 1 2. Differentiability and conformality 3 3. M¨obius transforms 7 4. Integration 12 5. Harmonic functions 19 6. The winding number 21 ... two polynomials (but the latter is needed only for the definition of the Riemann surfaces of an algebraic germ ...

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