**AMS Chelsea Publishing**

Volume: 371;
1973;
160 pp;
Hardcover

MSC: Primary 30;

Print ISBN: 978-0-8218-5270-5

Product Code: CHEL/371.H

List Price: $37.00

AMS Member Price: $33.30

MAA Member Price: $33.30

**Electronic ISBN: 978-1-4704-1578-5
Product Code: CHEL/371.H.E**

List Price: $37.00

AMS Member Price: $29.60

MAA Member Price: $33.30

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#### Supplemental Materials

# Conformal Invariants: Topics in Geometric Function Theory

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*Lars V. Ahlfors*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

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

#### Readership

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

#### Table of Contents

# Table of Contents

## Conformal Invariants: Topics in Geometric Function Theory

- Cover Cover11
- Title page iii4
- Contents v6
- Foreword ix10
- Preface xi12
- Applications of Schwarz’s lemma 114
- Capacity 2336
- Harmonic measure 3750
- Extremal length 5063
- Elementary theory of univalent functions 8295
- Löewner’s method 92105
- The Schiffer variation 98111
- Properties of the extremal functions 107120
- Riemann surfaces 125138
- The uniformization theorem 136149
- Bibliography 152165
- Index 156169
- Errata 159172
- Back Cover Back Cover1177