**University Lecture Series**

Volume: 58;
2012;
90 pp;
Softcover

MSC: Primary 30; 41; 47; 42; 46; 26; 11; 60;

Print ISBN: 978-0-8218-7559-9

Product Code: ULECT/58

List Price: $31.00

AMS Member Price: $24.80

MAA Member Price: $27.90

**Electronic ISBN: 978-0-8218-8489-8
Product Code: ULECT/58.E**

List Price: $31.00

AMS Member Price: $24.80

MAA Member Price: $27.90

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#### Supplemental Materials

# Complex Proofs of Real Theorems

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*Peter D. Lax; Lawrence Zalcman*

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

# Table of Contents

## Complex Proofs of Real Theorems

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Early triumphs 114 free
- Approximation 518
- Operator theory 1326
- Harmonic analysis 4558
- Banach algebras: The Gleason-Kahane-Żelazko theorem 6376
- Complex dynamics: The Fatou-Julia-Baker theorem 6780
- The prime number theorem 7184
- Coda: Transonic airfoils and SLE 7790
- Liouville’s theorem in Banach spaces 8194
- The Borel-Carathéodory inequality 8396
- Phragmén-Lindelöf theorems 8598
- Normal families 87100
- Back Cover Back Cover1106