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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. Problems58 chapter 4. Riemann surfaces : definitions, examples, basic properties631. The basic definitions632. Examples643. Functions on Riemann surfaces674.

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 7. Problems 24 Chapter 2. From zto the Riemann mapping theorem: some finer points of basic complex analysis 27 1.

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