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Complex Proofs of Real Theorems

Peter D. Lax Courant Institute, New York, NY
Lawrence Zalcman Bar-Ilan University, Ramat Gan, Israel
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Softcover ISBN: 978-0-8218-7559-9
Product Code: ULECT/58
List Price: $33.00 MAA Member Price:$29.70
AMS Member Price: $26.40 Electronic ISBN: 978-0-8218-8489-8 Product Code: ULECT/58.E List Price:$31.00
MAA Member Price: $27.90 AMS Member Price:$24.80
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List Price: $49.50 MAA Member Price:$44.55
AMS Member Price: $39.60 Click above image for expanded view Complex Proofs of Real Theorems Peter D. Lax Courant Institute, New York, NY Lawrence Zalcman Bar-Ilan University, Ramat Gan, Israel Available Formats:  Softcover ISBN: 978-0-8218-7559-9 Product Code: ULECT/58  List Price:$33.00 MAA Member Price: $29.70 AMS Member Price:$26.40
 Electronic ISBN: 978-0-8218-8489-8 Product Code: ULECT/58.E
 List Price: $31.00 MAA Member Price:$27.90 AMS Member Price: $24.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$49.50 MAA Member Price: $44.55 AMS Member Price:$39.60
• Book Details

University Lecture Series
Volume: 582012; 90 pp
MSC: Primary 30; 41; 47; 42; 46; 26; 11; 60;

Complex Proofs of Real Theorems is an extended meditation on Hadamard's famous dictum, “The shortest and best way between two truths of the real domain often passes through the imaginary one.” Directed at an audience acquainted with analysis at the first year graduate level, it aims at illustrating how complex variables can be used to provide quick and efficient proofs of a wide variety of important results in such areas of analysis as approximation theory, operator theory, harmonic analysis, and complex dynamics.

Topics discussed include weighted approximation on the line, Müntz's theorem, Toeplitz operators, Beurling's theorem on the invariant spaces of the shift operator, prediction theory, the Riesz convexity theorem, the Paley–Wiener theorem, the Titchmarsh convolution theorem, the Gleason–Kahane–Żelazko theorem, and the Fatou–Julia–Baker theorem. The discussion begins with the world's shortest proof of the fundamental theorem of algebra and concludes with Newman's almost effortless proof of the prime number theorem. Four brief appendices provide all necessary background in complex analysis beyond the standard first year graduate course. Lovers of analysis and beautiful proofs will read and reread this slim volume with pleasure and profit.

Graduate students and research mathematicians interested in analysis.

• Chapters
• Chapter 1. Early triumphs
• Chapter 2. Approximation
• Chapter 3. Operator theory
• Chapter 4. Harmonic analysis
• Chapter 5. Banach algebras: The Gleason-Kahane-Żelazko theorem
• Chapter 6. Complex dynamics: The Fatou-Julia-Baker theorem
• Chapter 7. The prime number theorem
• Coda: Transonic airfoils and SLE
• Appendix A. Liouville’s theorem in Banach spaces
• Appendix B. The Borel-Carathéodory inequality
• Appendix C. Phragmén-Lindelöf theorems
• Appendix D. Normal families

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 582012; 90 pp
MSC: Primary 30; 41; 47; 42; 46; 26; 11; 60;

Complex Proofs of Real Theorems is an extended meditation on Hadamard's famous dictum, “The shortest and best way between two truths of the real domain often passes through the imaginary one.” Directed at an audience acquainted with analysis at the first year graduate level, it aims at illustrating how complex variables can be used to provide quick and efficient proofs of a wide variety of important results in such areas of analysis as approximation theory, operator theory, harmonic analysis, and complex dynamics.

Topics discussed include weighted approximation on the line, Müntz's theorem, Toeplitz operators, Beurling's theorem on the invariant spaces of the shift operator, prediction theory, the Riesz convexity theorem, the Paley–Wiener theorem, the Titchmarsh convolution theorem, the Gleason–Kahane–Żelazko theorem, and the Fatou–Julia–Baker theorem. The discussion begins with the world's shortest proof of the fundamental theorem of algebra and concludes with Newman's almost effortless proof of the prime number theorem. Four brief appendices provide all necessary background in complex analysis beyond the standard first year graduate course. Lovers of analysis and beautiful proofs will read and reread this slim volume with pleasure and profit.

Graduate students and research mathematicians interested in analysis.

• Chapters
• Chapter 1. Early triumphs
• Chapter 2. Approximation
• Chapter 3. Operator theory
• Chapter 4. Harmonic analysis
• Chapter 5. Banach algebras: The Gleason-Kahane-Żelazko theorem
• Chapter 6. Complex dynamics: The Fatou-Julia-Baker theorem
• Chapter 7. The prime number theorem
• Coda: Transonic airfoils and SLE
• Appendix A. Liouville’s theorem in Banach spaces
• Appendix B. The Borel-Carathéodory inequality
• Appendix C. Phragmén-Lindelöf theorems
• Appendix D. Normal families
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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