Errata in Function Theory of One Complex Variable, second edition,
by Robert E. Greene and Steven G. Krantz
(list compiled by Harold P. Boas)
Math 618, Theory of Functions of a Complex Variable II, Spring 2006

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Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 15
Chapter 16
Appendix A

Chapter 8

page 258, line 8

In the fourth line of the first proof, the expression $e^{\left|a_j\right|}$ should be $e^{\left|a_j\right|/2}$ with a factor of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ in the exponent.

page 258, line -8

The stated inequality $P_N\le \exp M$ is valid, but not for the reason stated. The cited Corollary 8.1.3 contains the additional hypothesis that $\left|a_j\right|<1$, and this hypothesis does not hold in the present context.
One way to correct the argument is to observe that convergence depends only on what happens for large $j$. When $j$ is sufficiently large, one does have that $\left|a_j\right|<1$ because of the hypothesized convergence of the series $\sum _{j=1}^{\infty }\left|a_j\right|$.
A better way to correct the argument is to observe that the left-hand inequality in (8.1.2.1) holds for every non-negative value of $x$ (not just for $x$ less than $1$), so the right-hand inequality in Corollary 8.1.3 holds for arbitrary $\left|a_j\right|$ (not just for $\left|a_j\right|$ less than $1$). The observation is needed again on page 262 in the first paragraph of the proof of Theorem 8.1.9.

page 259, line 2

The stated inequality does not necessarily hold for all $N$, because the cited Corollary 8.1.3 contains the additional hypothesis that $\left|a_j\right|<1$.
Correcting the argument is simple. Indeed, it is easy to see that convergence of the infinite product $\prod _{j=1}^{\infty }\left(1+\left|a_j\right|\right)$ implies that $\underset{j\to \infty }{lim}\left|a_j\right|=0$. Therefore $\left|a_j\right|<1$ for all but finitely many values of $j$. Both for infinite products and for infinite series, convergence (or divergence) is unaffected by changing the value of a finite number of terms. Consequently, there is actually no loss of generality in assuming that $\left|a_j\right|<1$ for every value of $j$.

page 260, line 6

In the second sentence of the proof, the statement that $\underset{j\to \infty }{lim}\left|a_j\right|=0$ is valid, but the deduction of this statement from Corollary 8.1.5 is circular, because the statement is needed to fill the gap in the proof of Corollary 8.1.5. (See the preceding correction.)

page 262, lines -9 and -6

For $P_N$ and $P_M$ read $F_N$ and $F_M$. The function $P_N$ (introduced in the first paragraph of the proof) denotes the partial product $\prod _{j=1}^{\infty }\left(1+\left|f_j\right|\right)$ with absolute-value signs. What is wanted in the middle of the proof is the function $F_N$ from (8.1.9.2), the partial product without the absolute-value signs.

page 263, line 2

In the second product on the right-hand side of the display, the upper limit should be $N$ instead of $\infty$.

page 265, Theorem 8.2.2 and proof

The function denoted by an uppercase $F$ in the statement of the theorem is denoted by a lowercase $f$ in the proof.

page 265, line -4

In the first line of the proof, $a_k$ should be $a_m$, since later in the proof $m$ denotes the order of the zero at the origin.

page 267, line 7

For “bounary” read “boundary”.

page 267, line 18

For “let $K$ be a compact subset of $U$” read “let $K$ be a compact subset of $U\backslash \left\{\infty \right\}$”.
The difficulty here is that the normalizing linear fractional transformation has put the point at infinity inside the domain $U$, but the authors want to make a construction for $z$ in the finite plane. One needs to check separately that the constructed function is (or extends to be) holomorphic at infinity (which is stated in the last line of the proof as an exercise).

page 269, last line

One needs to know additionally that the box $q_p^{j\text{'}}$ is disjoint from $A_{j\text{'}-1}$(a property required in the construction at lines 4-5). Indeed, Figure 8.2 shows a situation in which this property is violated!
A possible fix is to observe that if the chosen point $z$ is unsuitable, then one can move a distance less than $4^{-j-1}$ to find a new $z$ that will serve.
I think, however, that a technically simpler way to carry out the proof is to select from ${𝒬}_j$ those boxes whose closure intersects both $U$ and the complement of $U$. Then choose from each such box one point in $U$, discarding any duplicate points. This set of points does not have the separation property of exercise 74 in Chapter 4; nonetheless, properties (1) and (2) of Lemma 8.3.2 are satisfied.

page 270, line 15

In the first line of the proof, for “property 2” read “property 1”.

page 271, Lemma 8.3.5

In the definition of $B\left(z\right)$, change the upper limit from $-K$ to $-1$, and delete the condition “$K\ge -1$”.

page 272, proof, line 4

The claim “without loss of generality” needs justification, because composing with an inversion does not preserve principal parts. One needs to check that composing a principal part with an inversion produces a principal part plus a constant; since the conclusion of the theorem does not see local additive constants, the method is valid.

page 273, proof of Lemma 8.3.7

The proof is correct, but based on what you know about Laurent series, the proof should be one line. Namely, the function $v\left(z\right)$ is simply the principal part of the Laurent series about $\alpha$ of the function $\left[e_0+e_1\left(z-\alpha \right)+\mathrm{...}+e_p{\left(z-\alpha \right)}^p\right]/g\left(z\right)$.

page 274, Theorem 8.3.8

The statement of the theorem is correct, but the statement is weaker than what the authors intended to say. What is left out of the statement is that the order of the pole of $m$ at ${\alpha }_j$ equals the order of the pole of $s_j$ at ${\alpha }_j$.
Perhaps a clearer way to state the conclusion is that the difference $m-s_j$ is holomorphic at ${\alpha }_j$ with a zero of order at least $1+N\left(j\right)$.

page 274, proof

There is a notational inconsistency with the symbol ${\tilde{s}}_j$. In lines 5-6 of the proof, this symbol is defined to be a partial sum of its own Taylor series, which does not make sense. In the displayed formula at line 5, one should delete the symbols “${\tilde{s}}_j\left(z\right)=$”.
Actually, the proof is inefficient because it first multiplies by a function with a high-order zero and then divides by a function with a high-order zero. Here is a simpler argument. By the Weierstrass theorem there is a holomorphic function $f$ with a zero of order $1+N\left(j\right)$ at each ${\alpha }_j$ and no other zeroes. By the version of the Mittag-Leffler theorem already proved (Theorem 8.3.6a), there is a meromorphic function $g$ that has poles only at the ${\alpha }_j$ and whose principal part at ${\alpha }_j$ agrees with the principal part of $s_j/f$. The product $f\times g$ does the job.

page 276, exercise 10

The first sentence of the exercise should conclude, “and no other zeroes”.

page 276, exercise 11

As in exercise 10, the intent is that the indicated zeroes are the only zeroes of the entire function.

pages 276-277, exercise 14

The wording is imprecise, for normal convergence is not something that happens pointwise. Presumably the intent is “find the largest open set on which the product converges normally”.

Chapter 9

page 281, line 4

The statement “(*) applies to $g$” is misleading, because equation (*) assumes that the function $g$ is defined in a neighborhood of the closed disc of radius $1$, but the function $g$ here is defined in a neighborhood of the closed disc of radius $r$. The authors are actually applying equation (*) to the function $z\mapsto g\left(rz\right)$.

page 281, line 12

The end-of-proof symbol is missing.

page 284, line 17

From here until the end of the proof on the next page, the symbol “$D$” is an abbreviation for “$D\left(0,1\right)$”. (This item was contributed by Jared Teslow.)

page 285, section 9.2, second paragraph, displayed equation

The strict inequality “$<$” should be replaced by weak inequality “$\le$” to accommodate the case when $z=0$. Strict inequality does hold when $z$ is any non-zero point in the open unit disc.

page 286, line -6

The strict inequality in the middle of the display is false when $z=0$. Since $\left|\psi \left(0\right)\right|=0<1$, this inaccuracy is trivial to repair.

page 286, line -4

The authors intend $\delta$ to be a suitable positive real number.

page 288, line -11

The intent of the statement “$f$ is of finite order if it grows exponentially” is “if it grows at most exponentially” (more precisely, $f$ grows no faster than the exponential of a polynomial in $\left|z\right|$).

page 289, proof of Lemma 9.3.1

The proof is not complete until one says why the assumption in the first line of the proof entails no loss of generality. The lacuna is easy to fill: approximate $r$ from above by a decreasing sequence of radii for which the assumption does hold, and observe that the desired inequality persists in the limit.

page 290, line 6

The right-hand side of the inequality should be ${\left(\left|a_j\right|+\delta \right)}^{\lambda +\epsilon }$ instead of ${\left|a_j\right|}^{\lambda +\epsilon }$. (That is what follows from the preceding displayed inequality, and it is the reason that the authors send $\delta$ to zero in the next line.)

page 290, line 8

The displayed inequality is correct, but essentially the same argument implies a stronger conclusion than the one stated in the theorem. Namely, replace $\lambda +1$ by $\lambda +\gamma$, and in line 9 set $\epsilon$ equal to $\gamma /2$. One deduces that for every positive $\gamma$ the series ${\sum }_j{\left|a_j\right|}^{-\left(\lambda +\gamma \right)}$ converges.
The authors actually need this stronger result in the very next paragraph!

page 290, lines 18 and 23

In the infinite products in the displayed equations, the subscript on the Weierstrass elementary factor $E$ should be $\left[\lambda \right]$ (greatest integer function) instead of $\lambda$.
In line 23, the argument of $E_{\lambda }$ is scrambled: it should be $z/a_n$, not $a/z_n$.

page 290, Lemma 9.3.3

The lemma is silent about whether $p$ is supposed to be an integer (a hypothesis that is made explicit in Lemma 9.3.4 and in Proposition 9.3.5). If $p$ is not an integer, then there is an ambiguity about what the $p\text{th}$ powers mean. Since the proof is based entirely on size estimates, however, the proof actually shows that ${lim}_{r\to \infty }{\sum }_{k=1}^{n\left(r\right)}{\left|{ā}_k\right|}^{p+1}{\left|r^2-{ā}_{k}z\right|}^{-p-1}=0$. Consequently, it does not matter whether $p$ is an integer or not.

page 291, proof of Lemma 9.3.4, first sentence

The restriction “when $r>2\left|z\right|$” is not relevant in this sentence. The restriction is used later in the proof to obtain the third line in formula (9.3.4.1) on page 292.

page 292, line 5

Since Jensen’s inequality was proved under the hypothesis that the function $f$ has no zeroes on the boundary of the disc of radius $r$, the indicated inequality has been established only for such radii.
A way to fill the gap is to take a limit along a sequence of good radii approaching $r$. Since the modulus of $f$ is bounded above on compact sets, one can apply Fatou’s lemma to bound the integral from below.

page 292, Proposition 9.3.5

It is implicit in the statement (and proof) of the proposition that the point $z$ is not equal to any of the zeroes $a_j$.
The hypothesis that $f$ is nonconstant is correct but redundant. If $f$ were constant, then the condition that $f\left(0\right)=1$ would imply that $f\left(z\right)$ is identically equal to $1$, in which case both sides of (9.3.5.1) trivially would be equal to $0$.

page 292, Proof

The first sentence should say “$2\left|z\right|<r$” instead of “$\left|z\right|<2r$”. The exercise cited in the next sentence is exercise 1 on page 296.

page 293, Lemma 9.3.6

As in Proposition 9.3.5, there is an implicit assumption that $z$ is not equal to any of the zeroes $a_j$.

page 293, line -9

For “${\left(p-1\right)}^{\text{st}}$” read “${\left(p+1\right)}^{\text{st}}$”. Also, “of degree $p$” should be “of degree at most $p$”.

page 293, line -2

The intent of “Differentiate” is “Take the logarithmic derivative of”.

page 294, lines 10-11

Surely “Lemma 9.3.4” is a misprint for “Lemma 9.3.6”.

page 294, line -12

The definition of “rank” is wrong. What is wanted is the least non-negative integer, not the least positive integer. For example, if $a_n=n^2$, then the rank equals $0$.

page 294, line -6

For “index” read “rank”.

page 295, Theorem 9.3.9

The hypothesis “$c$ lies in the image of $f$” is unnecessary and is not used in the proof. The proof shows, in particular, that the range of an entire function of finite non-integral order is the whole complex plane.

page 295, line 14

The “and likewise” statement is wrong. What is true (and this is all that is needed in the proof) is that the inequality in line 15 holds for some sequence of points $z$ tending to infinity.

page 295, line -7

Presumably “values(s)” is a misprint for “values”. There is no ambiguity about how many values the logarithm has: there are infinitely many, all differing by integral multiples of $2\pi i$.

page 296, exercise 1

The statement is correct but possibly misleading. Jensen’s formula is stated in Theorem 9.1.2 on page 280 for a disc of radius $r$, and the formula in this problem is stated for a disc of radius $r$, but the two values of $r$ are different from each other in the suggested method of proof. The given function $\phi$ maps the disc of radius $1$ into the disc of radius $r$, so one needs to apply Theorem 9.1.2 specialized to the case of the unit disc.

page 297, exercise 10

Presumably the intended assumption is that the sum converges absolutely. Since the points $a_n$ are not necessarily all inside the unit disc, the series could in principle converge without converging absolutely. It is not clear to me that there is a reasonable solution to the exercise in the case of conditional convergence.

page 297, exercise 11

There is a typographical error: the lower limit on the integral should be $a$ instead of $d$.

page 297, exercise 12

The hint is confusing: induction is an appropriate method for proving the second inequality. Notice also that the first inequality needs to be proved only when $\left|z\right|<1$. The source for the argument in this exercise apparently is the book Complex Analysis by Lars V. Ahlfors (page 209 of the third edition).

page 297, exercise 13

In part (a), for “$f\left(z\right)$” read “$f\left(x\right)$”.

Chapter 10

page 300, line 9

For “as series” read “as a series”.

page 306, last paragraph

The argument is correct, but a simpler way to show that the set $S$ is open would be to argue directly. Namely, if $t$ is a point of $\left[0,1\right]$ that belongs to $S$, and if $\epsilon$ and $\tilde{\epsilon }$ are the numbers from part (3) of Definition 10.2.1 corresponding to the two analytic continuations, then every point of $\left[0,1\right]$ whose distance from $t$ is less than the minimum of $\epsilon$ and $\tilde{\epsilon }$ belongs to $S$.

page 311, line 9

For “functional element” read “function element”.

page 311, line 22

The statement that the projection is two-to-one is correct only if one adjusts the definitions appropriately. The difficulty is that if $\left(f,U\right)$ is a function element, and $V$ is a disc concentric with $U$ but with smaller radius, then $\left(f,U\right)$ and $\left(f,V\right)$ have the same projection. To get around this difficulty, one has to view the function elements $\left(f,U\right)$ and $\left(f,V\right)$ as representing the same point of the Riemann surface. In other words, the Riemann surface consists not of function elements but of equivalence classes of function elements. Two function elements $\left(f,U\right)$ and $\left(g,V\right)$ are equivalent if $U$ and $V$ are concentric discs and $f=g$ on the intersection $U\cap V$. (The authors hinted at this equivalence relation in a different context on page 307 in the first paragraph following the conclusion of the proof.)
The standard name for such an equivalence class is “a germ of an analytic function”. One can think of a germ as being a convergent power series.

page 311, line -12

In view of the preceding comment, one has to adjust the definition of neighborhoods in the Riemann surface. Given a germ, take a representative $\left(f,U\right)$ of the germ and declare a neighborhood of the germ to be the collection of germs that have representatives $\left(f_p,U_p\right)$ for which $p\in U$ and $\left(f_p,U_p\right)$ is a direct analytic continuation of $\left(f,U\right)$.

page 313, line -16

For “counted clockwise” read “counterclockwise”.

page 316, lines 9-11

The argument is incomplete, for we do not know a priori that the hypotheses of the Schwarz reflection principle hold. A more convincing reason why the mappings are linear fractional is the one indicated in the penultimate paragraph on the same page: the biholomorphisms of the unit disc are known from Chapter 6 to be linear fractional, and the disc is mapped biholomorphically to the upper half-plane by a linear fractional map (the Cayley transform). Incidentally, you already know the result being derived if you have done exercise 8 on page 202 in Chapter 6.

page 318, Proposition 10.5.3

Replace “modular group” by “subgroup $\Gamma$ of the modular group”.

page 318, line -10

The symbol $e$, which seems not to have been defined in this context, denotes the identity element in the group — not the base of natural logarithms!

page 318, last line

The statement “$d\left(a-2b\right)=\pm 1$” is correct, being less restrictive than the statement at the beginning of the preceding line, but the authors probably meant to write “$a-2b=\pm 1$”.

page 319, part (c)

In lines 6-7 of part (c), the logic is faulty. The inverse map does not necessarily inherit the hypothesis that $c\ne 2d$, and therefore the claim “we could also apply this argument to $h^{-1}$” is not justified (without further argument).
What needs to be done to fix this is to consider another case parallel to part (b): namely, the case that $c=-2a$. Then the hypothesis of part (c) can be changed to “$c\ne 0$ and $c\ne 2d$ and $c\ne -2a$”.

page 320, line 1

In line 1, and also twice in the final sentence of the proof, $h^{\prime }$ is a misprint for $\tilde{h}$.

page 320, line -2

The parenthetical remark cites Carathéodory’s theorem (to be proved in a subsequent chapter), but the statement of Carathéodory’s theorem on page 391 in Chapter 13 assumes that the domains are bounded, and that hypothesis is violated here since the fundamental domain $\Gamma$ is not bounded.
One way to fix this is to make a preliminary linear fractional transformation that moves the point at infinity to a finite point. Then the image of the fundamental becomes bounded, and Carathéodory’s theorem can be applied.

page 321, paragraphs 3-4

The displayed equations in paragraph 3 are correct, but since the indicated quantities are real, the conjugations can be removed. Moreover, the conjugations should be removed, because the (unstated) point of the equations (without the conjugations) is that they imply continuity of the function $\lambda$ (defined in paragraph 4) at points of $U$ lying on the boundary of some image $h\left(W\right)$ of the fundamental domain $W$.

page 321, line -9

The statement “this set, in turn, equals ${\lambda }^{-1}\left(\left\{0\right\}\right)$” does not make sense, for $0$ is not in the range of $\lambda$, and (as we are trying to show) $\lambda$ does not extend continuously to the boundary. One can make sense of such a statement only after making the counter-factual hypothesis that $\lambda$ extends to a neighborhood of some boundary point — but this hypothesis has not been made at this point in the argument.
The argument can be fixed essentially by rearranging the sentences in the paragraph.

page 322, line 16

The word “mappings” should be “mapping” (singular).

page 326, line 5

The displayed formula has an undefined term when $m=n=0$. That term needs to be split off and considered separately. The same comment applies to lines 13, 16, and 22, as well as to line 3 on the next page.

page 326, line 7 (and from there to the end of the chapter)

The doubly periodic function of Weierstrass traditionally is denoted by a special typographical symbol ℘, which is available in TeX through the control sequence \wp and in HTML through the entity &weierp;.

page 328, lines 9 and 13

The constants $A_1$ and $A_3$ in line 9 change their identity in line 13. Thereafter, these two constants maintain their identity.

page 329, line 5

For “constants” read “constant” (singular).

page 330, Exercise 1

In line 4 of the exercise, after the initial word “and”, the left parenthesis is missing on the function element $\left(f_2,U_2\right)$.

page 332, Exercise 18

In lines 6-7 of the exercise, the words “irrational” and “rational” are interchanged.

Chapter 11

page 336, last paragraph

The definition of simple connectivity is stated only for domains (connected open sets), but one needs the definition for more general sets in exercises 4, 5, 10, 11, 14, 15, and 16 at the end of the chapter. As indicated in exercise 24 on page 355, the same definition makes sense for any set that is path connected.

page 338, line -11

The word “required” should be “require”.

page 340, line 3

The claim that the index is an integer needs justification. This was checked in Lemma 4.5.5 on page 124 when the curve is piecewise $C^1$, but in the present context the curve is only continuous.
One way to fill the gap is to show that the integral along a continuous curve can be arbitrarily well approximated by integrals along nearby piecewise $C^1$ curves.

page 341, Figure 11.1

The figure is not quantitatively correct, for the distance from the illustrated curve $\gamma$ to the complement of $U$ is less than twice the length $\delta$ of the sides of the squares.

page 341, line 8

Since $C_{\delta }$ is not necessarily the union of a collection of disjoint simple closed curves, the meaning of “their orientations are consistent” is unclear.
What is used in the proof is that each edge belonging to $C_{\delta }$ is an edge of exactly one square $Q_j$, and the orientation of the edge is chosen to be compatible with the usual (counterclockwise) orientation of the boundary of that square. This implies that (as usual) $U_{\delta }$ lies to the left of the oriented curve $C_{\delta }$.

page 342, line 7

The version of Fubini’s theorem in the appendix is not directly applicable, for according to the definition of integration along a continuous (but not differentiable) curve given on page 338 at the beginning of the section, the “integral” over $\gamma$ is not really an integral, but instead is a sum of values of anti-derivatives at points of a subdivision of the curve.
One way to justify the calculation is to show that the definition on page 338 actually does compute the integral along a piecewise $C^1$ curve that approximates $\gamma$.

page 346, line 7

The point of the argument is to give a concrete construction that does not depend on geometric intuition. In other words, one does not want to say, “obviously one can draw a curve in $U$ that encircles the point $a$”. Consequently, the claim “[t]his set of edges is clearly the union of a finite number of oriented, piecewise linear, closed curves” vitiates the proof.
The claim is correct, but to verify the claim requires a little combinatorics. One way to detect that a set of oriented edges constitutes a union of closed curves is that the sequence of initial points of the edges is a permutation of the sequence of terminal points of the edges. This property holds for the edges of a square, persists under taking unions, and continues to hold after deletion of a pair of edges common to two squares (but with opposite orientation).

page 348, Proposition 11.4.3

A missing punctuation mark obscures the meaning. In the third line of the proposition, insert a comma after “of $U$”.

page 349, line -11

A missing punctuation mark obscures the meaning. Insert a comma after “$k=0$”.

page 350, second displayed formula

The incorrectly sized parenthesis should cause no confusion.

page 354, Exercise 22

On page 335 at the beginning of the chapter, the convention was established that a product of curves is read left-to-right. In this exercise, that convention is reversed.

page 357, last two lines

The indicated continuity holds assuming that the “wrapping” is done in a continuous fashion. Notice that $\pi$ cannot be uniformly continuous, because there are arbitrarily close points of $G$ that map to opposite ends of a diameter of a unit circle.

page 358, exercise 30 part (g)

For “$R$ is restricted to $S$” read “$R$ restricted to $S$”.

Chapter 12

page 362, Figure 12.1

The set $K_k$ contains not only the two pieces indicated by arrows, but also a segment of the real axis.

page 363, Example 12.1.4

The set on which convergence to $0$ takes place is written incorrectly: “$x>0$” should be “$x\ge 0$”.
Actually more is true. Pointwise convergence to $0$ occurs on $ℂ\backslash \left\{0\right\}$, because every point other than the origin eventually is contained in $L_k$.

page 364, line -12

Since “cf.” abbreviates a single Latin word (confer, meaning “compare”), there should be no period after the “c”.

page 365, equation 12.1.5.2

The left-hand side needs a factor of ${\left(2\pi i\right)}^{-1}$. The mistake propagates to equation 12.1.5.3 on the next page.

page 366, Figure 12.4

The label $G$ in the diagram does not appear in the proof of Proposition 12.1.5. The proof does have a script letter $𝒢$, but that symbol denotes a set whose elements are boxes, while the $G$ in the figure is a point set that is equal to the union of the boxes.

page 366, Lemma 12.1.6

In line 2 of the lemma, the point $P$ should be in $ℂ\backslash K$, not $\widehat{ℂ}\backslash K$. The displayed formula in the statement of the lemma makes no sense if $P$ is the point at infinity (unless one interprets $1/\infty$ as 0).
The proof of Theorem 12.1.1 on the next page is unaffected, because if the pole is already at infinity, then there is no need to push the pole.

page 366, last sentence

Here one is assuming that $Q$ is not the point at infinity. (As indicated at the end of the proof on the next page, that case is left as an exercise for the reader.)
The inequality $\left|Q-Q^{\prime }\right|<dist\left(Q,K\right)$ is not sufficient to apply Lemma 8.3.5 directly. What is needed instead is the inequality $\left|Q-Q^{\prime }\right|<dist\left(Q^{\prime },K\right)$. Since one needs the right-hand side of the inequality to depend only on $Q$, a suitable inequality to make the proof work is $\left|Q-Q^{\prime }\right|<dist\left(Q,K\right)/2$.
On the other hand, in line 2 on page 367 the factor of 1/2 is not needed, and there the inequality $\left|Q-Q^{\prime }\right|<dist\left(Q,K\right)$ would suffice.

page 367, Proof

In line 5 of the proof, there is no need to invoke partial fractions, because the proof of Proposition 12.1.5 produced an expression of the indicated form, and indeed that expression is on the facing page in equation 12.1.5.3. One should, however, say that in the displayed equation for $\tilde{r}$, the numbers ${\alpha }_j$ are all non-zero, as this property is used twice in the proof.
In line 7 of the proof, the numbers ${\beta }_j$ should be in $ℂ\backslash K$ instead of $\widehat{ℂ}\backslash K$, for the subsequent display makes no sense if ${\beta }_j$ is the point at infinity.

page 369, line 4

The Note at line -6 probably belongs immediately after the statement of Lemma 12.2.2.

page 369, Lemma 12.2.3

The third sentence of the statement does not make sense, because $\zeta$ is quantified twice. Deleting “for each $\zeta \in D\left(P,r\right)$” would produce a correct sentence.
The statement of property 2 should carry the proviso that $z\ne \zeta$ and $z\ne \infty$. If one chooses to interpret the inequality in those two cases according to the convention that $1/0=\infty$ and $1/\infty =0$, then the strict inequality sign $<$ must be replaced with the weak inequality sign $\le$ (which indeed is the symbol that appears at the end of the proof at the bottom of the next page).
The constants 400 and 3300 in properties 1 and 2 are wrong but irrelevant. The point of writing explicit numbers instead of a symbolic constant “$C$” is to make absolutely clear that the constants do not depend on any of the various parameters in the problem. In tracing through the proof, you will see that at each step one can find an explicit numerical constant in each inequality. Ultimately one ends up with an upper bound of the form $C\times \epsilon$, with $C$ independent of $\epsilon$, and whether the constant $C$ is equal to 400 or 400,000 makes no difference.

page 370, line 5

Cauchy’s estimate is being applied to a function that is bounded by $8/r$ on a disc of radius $1/r$. Consequently, the coefficient of the quadratic term in the series expansion is bounded above by $8r$, not the claimed $4r$.

page 370, line 7

The numbers in the displayed inequality do not follow from the preceding discussion. Notice for future use that ${\phi }_{\zeta }$ is defined for every complex number $\zeta$, and from line 3 one has the estimate $\left|{\phi }_{\zeta }\left(z\right)\right|\le \left(8/r\right)+\left(\left|\zeta \right|+8r\right){\left(8/r\right)}^2$. Consequently, if $\zeta \in D\left(0,r\right)$, then $\left|{\phi }_{\zeta }\left(z\right)\right|\le 584/r$ for all $z$ in the complement of $E$.
A similar estimate holds for $\zeta$ in a larger disc. For instance, if $\zeta \in D\left(0,2r\right)$, then $\left|{\phi }_{\zeta }\left(z\right)\right|\le 648/r$.

page 370, line 14

The denominator in the fraction should be the number $2$, not the letter $z$.

page 370, line -7

Since the constant 400 was wrong earlier on the page, the constant 3300 is wrong too. The preceding considerations imply that $\left|\eta \left(z\right)\right|\le 4676r^2$ when $\zeta \in D\left(0,r\right)$ and $\left|\eta \left(z\right)\right|\le \mathrm{17,505}{r}^2$ when $\zeta \in D\left(0,2r\right)$.

page 372, line -12

In the penultimate display in the proof, the second integrand should have ${\lambda }_r$ instead of $\lambda r$.

page 374, Final Argument

In the first sentence, the claim about the locations of the centers is incorrect. Indeed, if $z$ is a point of $H$ at distance exactly $r$ from the complement of $K$, then (since the complement of $K$ is open) no disc of radius $r$ centered at a point of the complement of $K$ will contain the point $z$. For the covering argument to work, the discs must have radius slightly larger than $r$. As observed above, however, one has estimates analogous to those in Lemma 12.2.3 for $\zeta$ in a disc of radius $2r$, so there is no essential difficulty.
There is a gap in lines 5-7 of the argument. The intersection of the path with the closed disc of radius $r/2$ need not be connected, but the set $E$ in Lemma 12.2.3 is supposed to be connected. To fix this, take $E_j$ to be the part of the path from $P_j$ to the first point where the path hits the boundary of the disc $D\left(P_j,r/2\right)$.

page 375, line -11

In the denominator of the integral, the term ${\left(z-\zeta \right)}^3$ should have absolute-value signs.
In the same formula, there does not seem to be any special reason to use discs of radius $2r$ instead of radius $r$.
Also, writing the error term as $𝒪\left(\omega \left(r\right)\right)$ vitiates the goal of displaying explicit constants in the inequalities. This error term is actually $4\omega \left(r\right)r^{-1}{\int \int }_{H\cap D\left(z,2r\right)}{\left|z-\zeta \right|}^{-1}\mathrm{d\xi }\mathrm{d\eta }$. Integrating over the whole disc $D\left(z,2r\right)$ in polar coordinates centered at $z$ gives the upper bound $16\pi \omega \left(r\right)$.

page 375, line -7

The single integral sign should be a double integral sign.

page 376, line 5

The number 19602 is not supported by the authors’ discussion. Aside from the inaccuracies in the constants pointed out above, the authors have not quantified the $𝒪\left(\omega \left(r\right)\right)$ error term from the preceding page. As indicated above, however, it is possible to write explicit absolute constants in every inequality.

page 377

The definition of continuum in the middle of the page is convenient in context but not standard: ordinarily a continuum is not required to have connected complement.

page 378, line 19

Radius $4r$ is all right, but the natural radius that corresponds to the set-up in the proof on page 374 is $2r$. The point is that there exists some $j$ for which $D\left(P_j,r\right)\subset D\left(P,2r\right)$.

page 379, line 4

The statement that the “corollary is in fact Mergelyan’s formulation of his theorem” is historically misleading. The relevant paper (in Russian) is S. N. Mergelyan, Uniform approximations of functions of a complex variable, Uspekhi Matematicheskikh Nauk 7 (1952) 31-122. Mergelyan does give that formulation, but it is not one of his main results that he dignifies with the title “Theorem”. It is merely a special case that he mentions in passing as “one consequence of Theorem 4.4”.

page 379, line -5

For “there exists points” read “there exist points”.

page 381, Exercise 16

A hypothesis is missing from this theorem about removable singularities: the function needs to be assumed bounded.

page 381, Exercise 18

The meaning of “in this case Mergelyan’s theorem fails” is “in this case the conclusion of Mergelyan’s theorem fails”.

page 382, line 1

Of course the set $E$ should be a proper subset of the sphere (not the whole sphere itself).

Chapter 15

page 451, line 2

The natural logarithm function of a real number, denoted on the preceding page by “log”, is here denoted by “ln”. The same inconsistency occurs elsewhere (notably on page 470, where the statement of the Theorem uses both notations in the same sentence).

page 451, line -6

The statement that “the hypothesis $x>1$ was used to see that the boundary term in the integration by parts vanished” is misleading, for what is needed there is only that $x>0$. The assumption that $x>1$ was used implicitly in the claim that the function $f_n$ is continuous at 0. The monotone convergence theorem (in Appendix A, page 490) requires continuity only on the open half-line, however, so actually the assumption that $x>0$ will do for the whole proof.

page 453, line -13

The parenthetical remark “as long as the new series converges” is unnecessary, because by Corollary 3.5.2 on page 89, normal convergence is preserved under differentiation.

page 455, line 6

The parenthetical justification for interchanging the order of integration is inadequate, for the version of Fubini’s theorem stated in Appendix A (on page 489) covers only the case of continuous functions on bounded domains.
In the case at hand, the improper integrals are all absolutely convergent, so a more general version of Fubini’s theorem does apply.

page 455, Proposition 15.1.14

The hypothesis is wrong. One needs $z$ and $w$ to have positive real part in order for the integral to converge.

page 456, line 4

For “meromorphic” read “holomorphic”.

page 456, Proposition 15.2.1

The symbol $𝒫$ denoting the set of prime numbers is in one font in line 2 of the proposition and in another font in lines 3 and 4.

page 457, line -5

The given justification for interchanging the sum and the integral is inadequate, because when $Rez<2$ the sum does not converge uniformly. Indeed, the limit function is not a continuous function of $t$ at $0$.
One can justify the interchange by checking absolute convergence.

page 458, line 1

The indicated region is the complement of a closed half-plane. What was intended was the complement of a closed half-line.

page 458, line 2

The unidentified quantity $k$ is supposed to be an arbitrary positive integer.

page 459, line 3

Because one has taken the maximum over $\theta$, the factor $e^{-\theta Imz}$ should be $e^{\pi \left|Imz\right|}$.

page 459, line 9

The quantities ${\delta }^{\prime }$ and ${\delta }^{\prime \prime }$ (which are, in fact, equal to each other) are functions of $\delta$ (and implicitly of $\epsilon$), not of $t$. They are small positive quantities that tend to $0$ with $\delta$. Thus the statement “chosen so that $\left(-\pi +{\delta }^{\prime }\right)$ and $\left(\pi -{\delta }^{\prime \prime }\right)$ are the arguments of their initial/terminal points” is confusing at best.

page 459, line -6

The stated inequality is incorrect. First of all, the positive real axis is precisely the set where the function $u$ is not defined. Secondly, if one takes the limit of $\left|u\left(w\right)\right|$ as $w$ approaches the axis, one gets an additional factor of $\exp \left(\pi \left|Imz\right|\right)$ from the complex power of $-1$. (A similar error occurs on page 461, line 17.)
Nonetheless, one can bound $\left|u\left(w\right)\right|$ on the contour $C_{\epsilon }$ by a continuous, positive function of $z$ times ${\left|w\right|}^{Rez-1}{e}^{-\left|w\right|}$, so the conclusion that $H_{\epsilon }$ defines an entire function is correct.

page 460, proof of Proposition 15.2.4

The authors neglected to say that at the beginning of the proof, they took the limit as $\delta$ goes to $0$.

page 460, line -5

“It is enough to consider” is a correct reduction, but the proof requires in addition (page 461, line 18) that the real part of $z$ be negative.

page 461, line 18

The claim “it is apparent that” needs some justification. The (slightly wrong) inequality in line 17 implies that the contributions from the horizontal rays are easy to estimate (because of the exponential decay). To estimate the integral over the large circle requires bounding the denominator away from zero by a quantity that is independent of $n$.

page 462, lines 4-5

The reflection at issue is actually with respect to the line where the real part of $z$ is equal to $1/2$ (not $1$).

page 464, line 11

The statement “${\zeta }^4$ has a zero of order at least four there” is misleading. The function with the zero is not $z\mapsto \zeta \left(z\right)$ but $z\mapsto \zeta \left(z+i{t}_0\right)$.

page 464, line 16

In the denominator of the last summand in the displayed formula, the argument of $\zeta$ should be $x+2i{t}_0$ with a factor of 2 (as in the numerator).

page 464, line -7

The possessive form of “Riemann” should have the apostrophe preceding the “s”.

page 466, exercise 8

The letter $z$ in the exponent should be $t$.

page 466, exercise 12

Part (iv) is missing a limit as $k$ tends to infinity.

page 466, exercise 13

For “funciton” read “function”.

page 467, exercise 15

The displayed equation has two mistakes. The factor $e^w$ in the integrand should be $e^{-w}$ with a minus sign, and the factor $1/\left(2\pi i\right)$ should be deleted.

Chapter 16

page 470, first paragraph

The indicated numerical values are inaccurate. In lines 4 and 5, the given values for $Li\left(x\right)$ are too big by one unit, because $Li$ has been defined here to be the integral starting at $x=2$ rather than the principal-value integral starting at $x=0$ (which is the more common definition in the analysis literature). One can check the first value by the Maple command evalf(Li(10^6)-Li(2)); and the second value by the Mathematica command N[LogIntegral[10^9]-LogIntegral[2],20]. In line 6, the cited value of Meissel and Bertelsen for $\pi \left({10}^9\right)$ is well known to be wrong; the correct value of $\mathrm{50,847,534}$ is known to Mathematica via the command PrimePi[10^9].

page 470, line 20

For “Vallee” read “Vallée” with an acute accent on the first “e”. The same comment applies in line -13 and on page 479, line -6.

page 472, lines 1-3

The inequality in line 3 makes no sense if $s$ is a complex number, so in lines 1-2 one should replace “for all complex numbers $s$ with $Re\hspace{0.17em}s>1$” by “for all real numbers $s$ with $s>1$”.

page 472, line 6

The subscript on the limit should have the letter $s$, not the letter $x$.

page 477, item (1)

The stated inequalities are valid (for large $p$), but the second inequality is actually an equality; moreover, in view of item (2), the second inequality is not actually needed.

page 478, Proposition 16.2.2

The statement of the proposition is correct, but the conclusion in the second sentence is not what is needed in the argument coming up on page 480. What is needed there is that the function is holomorphic not just in a neighborhood of the point $1$, but in a neighborhood of the whole line where $\left\{Re\hspace{0.17em}z=1\right\}$. This conclusion is, in fact, what the proof of the proposition establishes.

page 479, line 6

In the parenthetical comment, $1+\alpha$ and $1-\alpha$ should be $1+i\alpha$ and $1-i\alpha$ with factors of $i$.

page 479, line 16

As shown correctly in the displayed equation in line 14, the sign between the two terms inside the fourth power should be a plus sign, not a minus sign.

page 480, line -11

We do not know that the integral is holomorphic in a neighborhood of the closed half-plane. Indeed, if we knew this, there would be no need for Proposition 16.2.4 on the next page. What we do know is that the function represented by the integral in the right half-plane has an analytic continuation to a neighborhood of the closed half-plane.

page 481, line 9

The statement that $\vartheta \left(t\right)e^{-t}$ tends to $0$ is not what we need to verify boundedness of the function $f$. Instead we need to know boundedness of $\vartheta \left(e^t\right)e^{-t}$, and this does follow from the inequality in the preceding line.

page 482, lines 5-6

In line 5, there should be a factor of $i$ multiplying $Im\hspace{0.17em}z$. Consequently, the constant that comes out in line 6 is actually $2$, not $4$, and one can write equality rather than inequality.
Possibly a simpler way to do the computation is to write $z=R{e}^{i\theta }$ and to observe that ${\displaystyle \left(1+\frac{z^2}{R^2}\right)\frac{1}{z}=\frac{1+e^{2i\theta }}{R{e}^{i\theta }}=\frac{2cos\theta }{R}=\frac{2Re\hspace{0.17em}z}{R^2}}$.

page 482, line 8

The indicated bound $4\pi B/R$ is correct, but what follows from the authors' previous inequality is the bound $2B/R$ (since the integral is multiplied by ${\left(2\pi i\right)}^{-1}$). The same comment applies to line 12. The constant in line 20 should be twice whatever constant is used in lines 8 and 12.

Appendix A

page 488, Green’s theorem, first sentence

The authors intended to assume additionally that the domain $\Omega$ is bounded.
The cited source (Rudin’s Principles of Mathematical Analysis) does not actually contain the stated version of Green’s theorem, for Rudin assumes that the boundary of the domain is class $C^2$. One can obtain the indicated version of Green’s theorem by exhausting the domain $\Omega$ by domains with class $C^2$ boundary and passing to the limit.

page 489, line 2

For “$\phi$” read “$f$”.

page 489, lines 9-10

The comment “we endow the two parts of $\partial {\Omega }_{\epsilon }$ with opposite orientations. Also notice that we do not write $\oint$ in the line integrals” is misleading. The authors have put in the minus sign by hand, so the two line integrals both have the standard orientation, and these integrals are complex line integrals in the sense of Definition 2.1.5 on page 32.

page 489, line 11

By “the second term” the authors mean “the second integral” (without the minus sign).

This document was last modified on 20 April 2006.