Complex analysis : fourth edition

Main subject categories: • Complex Analysis • Functions of a complex variableThe present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra reading material for students on their own. 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 recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: 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., DEs.The first part, I‒VIII, includes the basic properties of analytic functions, essentially what cannot be left out of, say, a one-semester course.The second and third parts of the book, IX‒XVI, deal with further assorted analytic aspects of. functions in many directions, which may lead to many other branches of analysis. The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read­ ing material for students on their own. A large number of routine exer­ cises 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 recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. Pt. 1. Basic Theory -- Ch. I. Complex Numbers And Functions -- Ch. Ii. Power Series -- Ch. Iii. Cauchy's Theorem, First Part -- Ch. Iv. Winding Numbers And Cauchy's Theorem -- Ch. V. Applications Of Cauchy's Integral Formula -- Ch. Vi. Calculus Of Residues -- Ch. Vii. Conformal Mappings -- Ch. Viii. Harmonic Functions -- Pt. 2. Geometric Function Theory -- Ch. Ix. Schwarz Reflection -- Ch. X. The Riemann Mapping Theorem -- Ch. Xi. Analytic Continuation Along Curves -- Pt. 3. Various Analytic Topics -- Ch. Xii. Applications Of The Maximum Modulus Principle And Jensen's Formula -- Ch. Xiii. Entire And Meromorphic Functions -- Ch. Xiv. Elliptic Functions -- Ch. Xv. The Gamma And Zeta Functions -- Ch. Xvi. The Prime Number Theorem -- App. 1. Summation By Parts And Non-absolute Convergence -- App. 2. Difference Equations -- App. 3. Analytic Differential Equations -- App. 4. Fixed Points Of A Fractional Linear Transformation -- App. 5. Cauchy's Formula For C[superscript Infinity] Functions -- App. 6. Cauchy's Theorem For Locally Integrable Vector Fields. Serge Lang. Includes Bibliographical References (p. [478]) And Index. Front Matter....Pages i-xiv Front Matter....Pages 1-1 Complex Numbers and Functions....Pages 3-36 Power Series....Pages 37-85 Cauchy’s Theorem, First Part....Pages 86-132 Winding Numbers and Cauchy’s Theorem....Pages 133-154 Applications of Cauchy’s Integral Formula....Pages 156-172 Calculus of Residues....Pages 173-207 Conformal Mappings....Pages 208-240 Harmonic Functions....Pages 241-290 Front Matter....Pages 291-291 Schwarz Reflection....Pages 293-305 The Riemann Mapping Theorem....Pages 306-321 Analytic Continuation Along Curves....Pages 322-336 Front Matter....Pages 337-337 Applications of the Maximum Modulus Principle and Jensen’s Formula....Pages 339-371 Entire and Meromorphic Functions....Pages 372-390 Elliptic Functions....Pages 391-407 The Gamma and Zeta Functions....Pages 408-439 The Prime Number Theorem....Pages 440-452 Back Matter....Pages 453-489 This is the fourth edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course. This is a revised edition, new examples and exercises have been added, and many minor improvements have been made throughout the text. "This is the fourth edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course."--BOOK JACKET

Now in its fourth edition, the first part of this book is devoted to the basic material of complex analysis, while the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than is found in other texts, and the resulting proofs often shed more light on the results than the standard proofs. While the first part is suitable for an introductory course at undergraduate level, the additional topics covered in the second part give the instructor of a gradute course a great deal of flexibility in structuring a more advanced course.

The complex numbers are a set of objects which can be added and multiplied, the sum and product of two complex numbers being also a complex number, and satisfy the following conditions. This well-established book covers the basic material of complex analysis, plus many special topics, such as the Riemann mapping theorem, the gamma function, and analytic continuation.