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Conformal Invariants: Topics in Geometric Function Theory

AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Available Formats:
Hardcover ISBN: 978-0-8218-5270-5
Product Code: CHEL/371.H
List Price: $40.00 MAA Member Price:$36.00
AMS Member Price: $36.00 Electronic ISBN: 978-1-4704-1578-5 Product Code: CHEL/371.H.E List Price:$37.00
MAA Member Price: $33.30 AMS Member Price:$29.60
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List Price: $60.00 MAA Member Price:$54.00
AMS Member Price: $54.00 Click above image for expanded view Conformal Invariants: Topics in Geometric Function Theory AMS Chelsea Publishing: An Imprint of the American Mathematical Society Available Formats:  Hardcover ISBN: 978-0-8218-5270-5 Product Code: CHEL/371.H  List Price:$40.00 MAA Member Price: $36.00 AMS Member Price:$36.00
 Electronic ISBN: 978-1-4704-1578-5 Product Code: CHEL/371.H.E
 List Price: $37.00 MAA Member Price:$33.30 AMS Member Price: $29.60 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:$60.00 MAA Member Price: $54.00 AMS Member Price:$54.00
• Book Details

AMS Chelsea Publishing
Volume: 3711973; 160 pp
MSC: Primary 30;

Most conformal invariants can be described in terms of extremal properties. Conformal invariants and extremal problems are therefore intimately linked and form together the central theme of this classic book which is primarily intended for students with approximately a year's background in complex variable theory. The book emphasizes the geometric approach as well as classical and semi-classical results which Lars Ahlfors felt every student of complex analysis should know before embarking on independent research.

At the time of the book's original appearance, much of this material had never appeared in book form, particularly the discussion of the theory of extremal length. Schiffer's variational method also receives special attention, and a proof of $\vert a_4\vert \leq 4$ is included which was new at the time of publication. The last two chapters give an introduction to Riemann surfaces, with topological and analytical background supplied to support a proof of the uniformization theorem.

Included in this new reprint is a Foreword by Peter Duren, F. W. Gehring, and Brad Osgood, as well as an extensive errata.

encompasses a wealth of material in a mere one hundred and fifty-one pages. Its purpose is to present an exposition of selected topics in the geometric theory of functions of one complex variable, which in the author's opinion should be known by all prospective workers in complex analysis. From a methodological point of view the approach of the book is dominated by the notion of conformal invariant and concomitantly by extremal considerations. …It is a splendid offering.

Reviewed for Math Reviews by M. H. Heins in 1975

Undergraduates, graduate students, and research mathematicians interested in geometric theory of functions of one complex variable.

• Chapters
• Chapter 1. Applications of Schwarz’s lemma
• Chapter 2. Capacity
• Chapter 3. Harmonic measure
• Chapter 4. Extremal length
• Chapter 5. Elementary theory of univalent functions
• Chapter 6. Löewner’s method
• Chapter 7. The Schiffer variation
• Chapter 8. Properties of the extremal functions
• Chapter 9. Riemann surfaces
• Chapter 10. The uniformization theorem

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Volume: 3711973; 160 pp
MSC: Primary 30;

Most conformal invariants can be described in terms of extremal properties. Conformal invariants and extremal problems are therefore intimately linked and form together the central theme of this classic book which is primarily intended for students with approximately a year's background in complex variable theory. The book emphasizes the geometric approach as well as classical and semi-classical results which Lars Ahlfors felt every student of complex analysis should know before embarking on independent research.

At the time of the book's original appearance, much of this material had never appeared in book form, particularly the discussion of the theory of extremal length. Schiffer's variational method also receives special attention, and a proof of $\vert a_4\vert \leq 4$ is included which was new at the time of publication. The last two chapters give an introduction to Riemann surfaces, with topological and analytical background supplied to support a proof of the uniformization theorem.

Included in this new reprint is a Foreword by Peter Duren, F. W. Gehring, and Brad Osgood, as well as an extensive errata.

encompasses a wealth of material in a mere one hundred and fifty-one pages. Its purpose is to present an exposition of selected topics in the geometric theory of functions of one complex variable, which in the author's opinion should be known by all prospective workers in complex analysis. From a methodological point of view the approach of the book is dominated by the notion of conformal invariant and concomitantly by extremal considerations. …It is a splendid offering.

Reviewed for Math Reviews by M. H. Heins in 1975

Undergraduates, graduate students, and research mathematicians interested in geometric theory of functions of one complex variable.

• Chapters
• Chapter 1. Applications of Schwarz’s lemma
• Chapter 2. Capacity
• Chapter 3. Harmonic measure
• Chapter 4. Extremal length
• Chapter 5. Elementary theory of univalent functions
• Chapter 6. Löewner’s method
• Chapter 7. The Schiffer variation
• Chapter 8. Properties of the extremal functions
• Chapter 9. Riemann surfaces
• Chapter 10. The uniformization theorem
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