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AuthorConway, John B. author
TitleFunctions of One Complex Variable [electronic resource] / by John B. Conway
ImprintNew York, NY : Springer US, 1973
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Descript XIII, 313 p. online resource


This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - I) arguments. The actual preยญ requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differยญ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Matheยญ matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc


I. The Complex Number System -- ยง1. The real numbers -- ยง2. The field of complex numbers -- ยง3. The complex plane -- ยง4. Polar representation and roots of complex numbers -- ยง5. Lines and half planes in the complex plane -- ยง6. The extended plane and its spherical representation -- II. Metric Spaces and the Topology of C -- ยง1. Definition and examples of metric spaces -- ยง2. Connectedness -- ยง3. Sequences and completeness -- ยง4. Compactness -- ยง5. Continuity -- ยง6. Uniform convergence -- III. Elementary Properties and Examples of Analytic Functions -- ยง1. Power series -- ยง2. Analytic functions -- ยง3. Analytic functions as mappings, Mรถbius transformations -- IV. Complex Integration -- ยง1. Riemann-Stieltjes integrals -- ยง2. Power series representation of analytic functions -- ยง3. Zeros of an analytic function -- ยง4. Cauchyโ{128}{153}s Theorem -- ยง5. The index of a closed curve -- ยง6. Cauchyโ{128}{153}s Integral Formula -- ยง7. Counting zeros; the Open Mapping Theorem -- ยง8. Goursatโ{128}{153}s Theorem -- V. Singularities -- ยง1. Classification of singularities -- ยง2. Residues -- ยง3. The Argument Principle -- VI. The Maximum Modules Theorem -- ยง1. The Maximum Principle -- ยง2. Schwarzโ{128}{153}s Lemma -- ยง3. Convex functions and Hadamardโ{128}{153}s Three Circles Theorem -- ยง4. Phragmen-Lindelรถf Theorem -- VII. Compactness and Convergence in the Space of Analytic Functions -- ยง1. The space of continuous functions C(G,?) -- ยง2. Spaces of analytic functions -- ยง3. Spaces of meromorphic functions -- ยง4. The Riemann Mapping Theorem -- ยง5. Weierstrass Factorization Theorem -- ยง6. Factorization of the sine function -- ยง7. The gamma function -- ยง8. The Riemann zeta function -- VIII. Rungeโ{128}{153}s Theorem -- ยง1. Rungeโ{128}{153}s Theorem -- ยง2. Another version of Cauchyโ{128}{153}s Theorem -- ยง3. Simple connectedness -- ยง4. Mittag-Lefflerโ{128}{153}s Theorem -- IX. Analytic Continuation and Riemann Surfaces -- ยง1. Schwarz Reflection Principle -- ยง2. Analytic Continuation Along A Path -- ยง3. Mondromy Theorem -- ยง4. Topological Spaces and Neighborhood Systems -- ยง5. The Sheaf of Germs of Analytic Functions on an Open Set -- ยง6. Analytic Manifolds -- ยง7. Covering spaces -- X. Harmonic Functions -- ยง1. Basic Properties of harmonic functions -- ยง2. Harmonic functions on a disk -- ยง3. Subharmonic and superharmonic functions -- ยง4. The Dirichlet Problem -- ยง5. Greenโ{128}{153}s Functions -- XI. Entire Functions -- ยง1. Jensenโ{128}{153}s Formula -- ยง2. The genus and order of an entire function -- ยง3. Hadamard Factorization Theorem -- XII. The Range of an Analytic Function -- ยง1. Blochโ{128}{153}s Theorem -- ยง2. The Little Picard Theorem -- ยง3. Schottkyโ{128}{153}s Theorem -- ยง4. The Great Picard Theorem -- Appendix: Calculus for Complex Valued Functions on an Interval -- List of Symbols

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