##### 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

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