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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
Complex Proofs of Real Theorems
Softcover ISBN:  978-0-8218-7559-9
Product Code:  ULECT/58
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-0-8218-8489-8
Product Code:  ULECT/58.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-0-8218-7559-9
eBook: ISBN:  978-0-8218-8489-8
Product Code:  ULECT/58.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
Complex Proofs of Real Theorems
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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
Softcover ISBN:  978-0-8218-7559-9
Product Code:  ULECT/58
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-0-8218-8489-8
Product Code:  ULECT/58.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN:  978-0-8218-7559-9
eBook ISBN:  978-0-8218-8489-8
Product Code:  ULECT/58.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
  • 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.

    Readership

    Graduate students and research mathematicians interested in analysis.

  • Table of Contents
     
     
    • 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 publishers of book reviews
    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.

Readership

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 publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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