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TitleThe Cauchy transform, potential theory, and conformal mapping
Author
LanguageEnglish
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Total Pages221
Table of Contents
                            Front Cover
Contents
Preface
Table of symbols
Chapter 1 - Introduction
Chapter 2 - The improved Cauchy integral formula
Chapter 3 - The Cauchy transform
Chapter 4 - The Hardy space, the Szegő projection, and the Kerzman-Stein formula
Chapter 5 - The Kerzman-Stein operator and kernel
Chapter 6 - The classical definition of the Hardy space
Chapter 7 - The Szegő kernel function
Chapter 8 - The Riemann mapping function
Chapter 9 - A density lemma and consequences
Chapter 10 - Solution of the Dirichlet problem in simply connected domains
Chapter 11 - The case of real analytic boundary
Chapter 12 - The transformation law for the Szegő kernel under conformal mappings
Chapter 13 - The Ahlfors map of a multiply connected domain
Chapter 14 - The Dirichlet problem in multiply connected domains
Chapter 15 - The Bergman space
Chapter 16 - Proper holomorphic mappings and the Bergman projection
Chapter 17 - The Solid Cauchy transform
Chapter 18 - The classical Neumann problem
Chapter 19 - Harmonic measure and the Szegő kernel
Chapter 20 - The Neumann problem in multiply connected domains
Chapter 21 - The Dirichlet problem again
Chapter 22 - Area quadrature domains
Chapter 23 - Arc length quadrature domains
Chapter 24 - The Hilbert transform
Chapter 25 - The Bergman kernel and the Szegő kernel
Chapter 26 - Pseudo-local property of the Cauchy transform and consequences
Chapter 27 - Zeroes of the Szegő kernel
Chapter 28 - The Kerzman-Stein integral equation
Chapter 29 - Local boundary behavior of holomorphic mappings
Chapter 30 - The dual space of A∞(Ω)
Chapter 31 - The Green’s function and the Bergman kernel
Chapter 32 - Zeroes of the Bergman kernel
Chapter 33 - Complexity in complex analysis
Chapter 34 - Area quadrature domains and the double
Appendix A - The Cauchy-Kovalevski theorem for the Cauchy-Riemann operator
Bibliographic Notes
Bibliography
Back Cover
                        
Document Text Contents
Page 1

The Cauchy Transform, Potential Theory and Conformal Mapping
explores the most central result in all of classical function theory, the
Cauchy integral formula, in a new and novel way based on an advance
made by Kerzman and Stein in 1976.

The book provides a fast track to understanding the Riemann Mapping
Theorem. The Dirichlet and Neumann problems for the Laplace opera-
tor are solved, the Poisson kernel is constructed, and the inhomog-
enous Cauchy-Reimann equations are solved concretely and efficiently
using formulas stemming from the Kerzman-Stein result.

These explicit formulas yield new numerical methods for computing the
classical objects of potential theory and conformal mapping, and the
book provides succinct, complete explanations of these methods.

Four new chapters have been added to this second edition: two on
quadrature domains and another two on complexity of the objects of
complex analysis and improved Riemann mapping theorems.

The book is suitable for pure and applied math students taking a begin-
ning graduate-level topics course on aspects of complex analysis as
well as physicists and engineers interested in a clear exposition on a
fundamental topic of complex analysis, methods, and their application.

K25868

w w w . c r c p r e s s . c o m

The Cauchy Transform,
Potential Theory
and Conformal
Mapping
2nd Edition

The Cauchy Transform, Potential Theory
and Conformal Mapping
2nd Edition

T
he C

au
chy T

ran
sform

, P
oten

tial T
heory an

d C
on

form
al M

appin
g

Steven R. Bell

B
ell

2nd
Edition

Mathematics

K25868_cover.indd 1 10/9/15 11:27 AM

Page 110

96 The Cauchy Transform, Potential Theory, and Conformal Mapping

Proof of the lemma. Divide the equation in the statement of the lemma
by La, and use identity (7.1) to see that our problem is equivalent to
proving that

ψ

La
= P (ψ/La ) + i P (ψ/Sa )T .

This is an orthogonal sum. The holomorphic part is obviously what it
should be. The proof of the lemma will be finished if we show that

P⊥
(
ψ

La

)
= i P (ψ/Sa )T .

Since P⊥u = P (uT )T , this is equivalent to showing that

P

(
ψ

La
T

)
= −iP (ψ/Sa ),

and this follows directly from identity (7.1). The proof of the lemma is
done and therefore, so is the proof of the theorem.

Page 111

19

Harmonic measure and the Szegő kernel

The expression −ih′T + iH ′T appears as the normal derivative of h+H
in the proof of Theorem 18.1. In a simply connected domain, we will
see that the condition −ih′T + iH ′T = 0 forces h′ ≡ 0 and H ′ ≡ 0.
Before we treat the Neumann problem in a multiply connected domain,
we must determine which functions h and H in H2(bΩ) satisfy the con-
dition −ihT + iHT = 0, i.e., satisfy hT = HT . By Theorem 4.3, a
function u that is expressible as both hT and HT would be orthogo-
nal to H2(bΩ) and orthogonal to conjugates of functions in H2(bΩ). We
will see that such functions are closely related to the classical harmonic
measure functions (which we define below).

In this chapter, we will be dealing with a bounded n-connected do-
main Ω with C∞ smooth boundary curves. Let {γj}nj=1 denote the
n boundary curves of Ω. For convenience, we may assume that γn is
the outer boundary curve, i.e., the one that also bounds the unbounded
component of the complement of Ω in C. The harmonic measure func-
tions {ωj}nj=1 associated to Ω are defined as follows. The function ωj is
equal to the harmonic function on Ω that solves the Dirichlet problem
with boundary data equal to one on γj and equal to zero on the other
boundary curves. Note that we know that these harmonic measure func-
tions are in C∞(Ω) by Theorem 14.2.

An important holomorphic function associated to ωj is the function
F ′j = 2∂ωj/∂z. The prime in the notation F


j is traditional; F


j is the

derivative of a multi-valued holomorphic function. To see this, suppose
that v is a harmonic conjugate for ωj on a small disc contained in Ω
and define Fj = ωj + iv there. By the Cauchy-Riemann equations, F


j =

(∂/∂x)ωj−i(∂/∂y)ωj = 2∂ωj/∂z. Thus, F ′j is the derivative of the multi-
valued function obtained by analytically continuing around Ω a germ of
ωj + iv where v is a local harmonic conjugate for ωj .

Our next theorem will show how the functions F ′j are related to the
Szegő kernel and its zeroes. Let a ∈ Ω be a given point. By Theorem 13.1,
the function Sa(z) = S(z, a) has exactly n− 1 zeroes in Ω (counted with
multiplicity). We will prove in Chapter 27 that the zeroes of S(z, a)
become simple zeroes as a approaches the boundary of Ω. We assume

97

Page 220

206 Bibliography

[K-T] N. Kerzman and M. Trummer Numerical conformal mapping
via the Szegő kernel, in Numerical conformal mapping, L. N.
Trefethen, ed., North Holland, Amsterdam, 1986, 111–123.

[Ko1] B. Korenblum An extension of the Nevanlinna theory, Acta
Math. 135 (1975), 187–219.

[Ko2] , A Beurling-type theorem, Acta Math. 138 (1976), 265–
293.

[Ne] Z. Nehari Conformal mapping, Dover, New York, 1952.

[Ru1] W. Rudin Principles of mathematical analysis, McGraw Hill,
New York, 1953.

[Ru2] , Real and complex analysis, McGraw Hill, New York,
1966.

[Sch] M. Schiffer Various types of orthogonalization, Duke Math. J.
17 (1950), 329–366.

[Sh] H. S. Shapiro The Schwarz function and its generalization to
higher dimensions, Univ. of Arkansas Lecture Notes in the
Mathematical Sciences, Wiley, New York, 1992.

[S-U] H. Shapiro, and C. Ullemar, Conformal mappings satisfying cer-
tain extremal properties and associated quadrature identities,
Research Report TRITA-MAT-1986-6, Royal Inst. of Technol-
ogy, 40 pp., 1981.

[Sp] M. Spivak Calculus on manifolds, Benjamin, New York, 1965.

[St] E. M. Stein Boundary behavior of holomorphic functions,
Princeton University Press, Princeton, 1972.

[Su-Ya] N. Suita and A. Yamada On the Lu Qi-Keng conjecture, Proc.
Amer. Math. Soc. 59 (1976), 222–224.

[T-W] B. A. Taylor and D. L. Williams Ideals in rings of analytic
functions with smooth boundary values, Canad. J. Math. 22
(1970), 1266–1283.

[Tr] M. Trummer An efficient implementation of a conformal map-
ping method based on the Szegő kernel, SIAM J. of Numer.
Anal. 23 (1986), 853–872.

[Yo1] K. Yosida Lectures on differential and integral equations, Wiley,
New York, 1960.

[Yo2] , Functional Analysis, Springer-Verlag, New York, 1980.

Page 221

The Cauchy Transform, Potential Theory and Conformal Mapping
explores the most central result in all of classical function theory, the
Cauchy integral formula, in a new and novel way based on an advance
made by Kerzman and Stein in 1976.

The book provides a fast track to understanding the Riemann Mapping
Theorem. The Dirichlet and Neumann problems for the Laplace opera-
tor are solved, the Poisson kernel is constructed, and the inhomog-
enous Cauchy-Reimann equations are solved concretely and efficiently
using formulas stemming from the Kerzman-Stein result.

These explicit formulas yield new numerical methods for computing the
classical objects of potential theory and conformal mapping, and the
book provides succinct, complete explanations of these methods.

Four new chapters have been added to this second edition: two on
quadrature domains and another two on complexity of the objects of
complex analysis and improved Riemann mapping theorems.

The book is suitable for pure and applied math students taking a begin-
ning graduate-level topics course on aspects of complex analysis as
well as physicists and engineers interested in a clear exposition on a
fundamental topic of complex analysis, methods, and their application.

K25868

w w w . c r c p r e s s . c o m

The Cauchy Transform,
Potential Theory
and Conformal
Mapping
2nd Edition

The Cauchy Transform, Potential Theory
and Conformal Mapping
2nd Edition

T
he C

au
chy T

ran
sform

, P
oten

tial T
heory an

d C
on

form
al M

appin
g

Steven R. Bell

B
ell

2nd
Edition

Mathematics

K25868_cover.indd 1 10/9/15 11:27 AM

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