Complex Analysis

1. Front Matter

   Pages i-xiv

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2. Basic Theory

   1. Front Matter

      Pages 1-1

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   2. Complex Numbers and Functions

      • Serge Lang

      Pages 3-37

   3. Power Series

      • Serge Lang

      Pages 38-86

   4. Cauchy’s Theorem, First Part

      • Serge Lang

      Pages 87-122

   5. Cauchy’s Theorem, Second Part

      • Serge Lang

      Pages 123-143

   6. Applications of Cauchy’s Integral Formula

      • Serge Lang

      Pages 144-164

   7. Calculus of Residues

      • Serge Lang

      Pages 165-195

   8. Conformal Mappings

      • Serge Lang

      Pages 196-223

   9. Harmonic Functions

      • Serge Lang

      Pages 224-251

3. Various Analytic Topics

   1. Front Matter

      Pages 253-253

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   2. Applications of the Maximum Modulus Principle

      • Serge Lang

      Pages 255-275

   3. Entire and Meromorphic Functions

      • Serge Lang

      Pages 276-291

   4. Elliptic Functions

      • Serge Lang

      Pages 292-306

   5. Differentiating Under an Integral

      • Serge Lang

      Pages 307-323

   6. Analytic Continuation

      • Serge Lang

      Pages 324-339

   7. The Riemann Mapping Theorem

      • Serge Lang

      Pages 340-358

4. Back Matter

   Pages 359-370

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The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. Somewhat more material has been included than can be covered at leisure in one term, to give opportunities for the instructor to exercise his taste, and lead the course in whatever direction strikes his fancy at the time. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recom­ mend to anyone to look through them. More recent texts have empha­ sized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex anal­ ysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. The systematic elementary development of formal and convergent power series was standard fare in the German texts, but only Cartan, in the more recent books, includes this material, which I think is quite essential, e. g. , for differential equations. I have written a short text, exhibiting these features, making it applicable to a wide variety of tastes. The book essentially decomposes into two parts.

• Analysis
• Complex analysis
• Meromorphic function
• calculus
• complex number
• differential equation
• elliptic function
• integral
• maximum
• real analysis
• residue

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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