Comments on Chapter 5

Page 1

In Section 5.1.1, line 2 of the first paragraph and line 1 of the second paragraph, for \({\mathbb{C}}(\Omega)\) read \(C(\Omega)\) with the letter \(C\) in regular mathematics font (not blackboard bold).

Page 2

In the last paragraph preceding Section 5.1.3, line 2, for \({\mathbb A}(\Omega)\) read \(A(\Omega)\) with the letter \(A\) in ordinary mathematics font (not blackboard bold).

Page 3

The concept termed bounded in Definition 5.1.6 is what most authors would call locally bounded.

Page 4

In the final displayed equation in the proof of Theorem 5.1.8, the term \( (r/2)^2\) in the denominator is correct but can be replaced by the sharper expression \(r^2/2\) (which is the expression that most naturally comes out of the argument, since \(|w-z_0|=r\)).

Page 5

In 5.1.10, line 2, for “om” read “on”.

In the first Remark, line 1, for \(C(\Omega\) read \(C(\Omega)\).

In the proof of 5.1.10, line 2, for \(f_n\}\) read \(\{f_n\}\).

Page 7

In Problem 6, line 2, delete the unmatched closing bracket.

Page 8

In Problem 7, the region at issue in Figure 5.1.1 is commonly called a Stolz angle. The statement of the problem is a version of what is known as the Lindelöf principle.

In Section 5.2, end of the first paragraph, the name Fejér is missing an accent.

Four lines from the bottom of the page, for \([h(\Omega] \cap [-h(\Omega]\) read \([h(\Omega)]\cap[-h(\Omega)].\)

Page 15

First line of the proof of Lemma 5.2.10, capitalize the word “let”.

In the proof of Lemma 5.2.11, line 6, for \(R=R_1+R_2\) read \(R=R_1+R_0\).

Page 22

Eight lines from the bottom of the page, \(C(0,r)\) should be \(C(z_0,r).\)

Five lines from the bottom of the page, preceding the period at the end of the line, the right-hand curly brace should be a right-hand parenthesis.

Page 23

At line 3, it should be mentioned that case (2) from the bottom of page 21 can be handled by a completely parallel argument in the sector \(d0cd.\)

The hypothesis in Lemma 5.3.7 should say that \(g\) is a one-to-one conformal map. (According to the authors' definition, a conformal map need be only locally one-to-one.) If \(g\) is not globally one-to-one, then the indicated integral computes the area of the image counting multiplicity. For instance, the function \(z^3\) maps the punctured unit disk onto the punctured unit disk with multiplicity three, and the integral computes the value \(3\pi.\)

Page 24

The hypothesis of Theorem 5.3.8 should say “conformal equivalence” instead of “conformal map”: the map is supposed to be globally one-to-one.

In the final line of formula (2), the closing right-hand parenthesis is displaced vertically: \(g'(re^{i\theta)}\) should be \(g'(re^{i\theta}).\)

Page 25

In the proof of Theorem 5.3.9, there is a spurious extra period at the end of the first sentence.

Harold P. Boas