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Introduction to Complex Analysis ... - Michael E. Taylor

Introduction to Complex AnalysisMichael Taylor12 ContentsChapter 1. Basic calculus in the Complex domain0. Complex numbers, power series, and exponentials1. Holomorphic functions, derivatives, and path integrals2. Holomorphic functions de ned by power series3. Exponential and trigonometric functions: Euler's formula4. Square roots, logs, and other inverse functionsI. 2is irrationalChapter 2. Going deeper { the Cauchy integral theorem and consequences5. The Cauchy integral theorem and the Cauchy integral formula6. The maximum principle, Liouville's theorem, and the fundamental theorem of al-gebra7. Harmonic functions on planar regions8. Morera's theorem, the Schwarz re ection principle, and Goursat's theorem9. In nite products10. Uniqueness and analytic continuation11. Singularities12. Laurent seriesC. Green's theoremF. The fundamental theorem of algebra (elementary proof)L.}

The text ends with a short collection of appendices. Some of these survey background material that the reader might have seen in an advanced calculus course, including material on convergence and compactness, and fftial calculus of several variables. Others develop tools that prove useful in the text, the Laplace asymptotic method, the Stieltjes

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