Comments on Chapter 4
- Page 3
- In the third line, two symbols are missing. The expression at the
end of the line should be \( \{ z: |z-z_0|\gt r_1 \}\).
- Page 5
- There is an extra closing right-hand parenthesis at the end of
the first formula.
- Page 7
- In item (b), the assumption “and \(n(\gamma,z)=0\)
for every \(z\notin D(z_0,\rho)\)” is automatically
satisfied by 3.4.4.
- Page 9
- In 4.2.4, line 3, insert a space following the comma.
- Page 10
- In the first line of 4.2.8, the hypothesis “neither \(f\)
nor \(g\) identically equal to zero on any component of
\(\Omega\)” is superfluous, in view of the strict
inequality (1).
- Three lines from the bottom of the page, notice that the pole
set \(P(f)\) is empty (by hypothesis).
- Page 11
- In Figure 4.2.1, the point labeled “i”
should be labeled “1”.
- Page 12
- In Figure 4.2.2, the label -5-21 should be -5-2i.
- Page 16
- In the first line of the proof of part (b), for \(n(f\circ\gamma)\) read \(n(f\circ\gamma,w)\).
- In both displayed formulas in 4.3.3, the differential \(dw\) should be \(dz\).
- Page 20
- In 4.5.2, first display, middle term, the delimiters in the denominator are wrongly nested. The left-hand and right-hand square brackets in the denominator can be deleted.
- Last paragraph preceding the Problems, line 3, read “linear fractional transformations” (plural).
- Problem 1, the subject–verb agreement is wrong. The end of the sentence should read “are again linear fractional transformations.”
- Page 21
- Problem 5(g) has a spurious closing right-hand parenthesis following \(\text{Res}(f,z_0)\).
- Page 23
- In the third line of the three-line display, delete the spurious closing right-hand parenthesis in the denominator.
- In Theorem 4.6.4, first line, capitalize “Then”.
- Page 26
- In the first line of 4.7.3, capitalize “Then”.
- Page 32
- In the first displayed formula in 4.8.1, there is a missing
closing right-hand parenthesis in the integrand:
\(\ln|f(Re^{it}|\) should be \(\ln|f(Re^{it})|.\)
- Page 33
- Four lines above 4.8.3, for “à priori” read
“a priori”. (The term “a priori” is
Latin, not French.)
- Page 34
- In the second displayed formula, the sum should run from
\(n+1\) to \(m,\) not \(1\) to \(n.\)
Similarly, in the line after the fourth displayed
equation, the limits on the product should be \(n+1\) to
\(m\) instead of \(1\) to \(n.\)
- In the third displayed equation, the sum needs a factor of \(k_j\).
[Contributed by Martijn de Vries, 2017.]
- Three lines above 4.8.4, insert a closing right-hand
parenthesis after the formula \(|a|\lt R\).
- In the line preceding 4.8.4, insert a terminal period.
- Page 35
- Seven lines from the bottom of the page, the claim that
\(P_z(-t)=P_z(t)\) is wrong: actually, \(P_z(-t) = P_{\bar z}(t)\). Nonetheless, the proof goes through with minor changes, for \(P_z(t)\) and \(P_{\bar z}(t)\) admit the same upper bound
\(\dfrac{1+|z|}{1-|z|}\) at line 2 on page 36.
[Contributed by Martijn de Vries, 2017.]
- Page 36
- In line 4, insert a closing right-hand parenthesis after “Problem 2”.
- Four lines from the end of the proof, “uniformly in
\(r\)” should say “uniformly in \(r\) in a bounded
interval, say \((1,2)\)”.
- Page 39
- In the last line of the proof of Lemma 4.9.4, insert a
closing right-hand parenthesis: \(f_{1}(D(0,r)\) should be \(f_{1}(D(0,r)).\)
- Page 40
- In the parenthetical remark preceding the third display, the
subject and verb do not agree; replace “repetition of
powers” with “repeated powers”.
- Page 41
- In 4.9.6, second paragraph, second line, insert a space after
the word “chain”.
- Page 42
- At line 8, insert an opening left-hand parenthesis to make
the formula read \( (g_j,E_j).\)
- In Definition 4.9.9, line 5, for “\(a\le t\le
t\)” read “\(a\le t\le b\)”.
- Page 43
- In the first proof, line 4, insert a less-than sign between the
ellipsis dots and \(t_n.\)
- Page 44
- In the final paragraph, line 3, for
“\(\gamma_{(}t_{n})\)” read “\(\gamma(t_{n})\)”.
Harold P. Boas