Additions, Changes, and Corrections
for
Functions of One Complex Variable
( Second edition, fourth printing)
by
John B Conway
This is a list of additions, changes, and corrections for my book Functions of One Complex Variable
(Second Edition,Fourth Printing). These corrections also apply to the fifth and sixth printing. The book is
currently in the seveth printing of the second edition and almost all of these corrections have been executed
there.
I have a separate list of additions and changes that will appear in the next edition. This is also available
from my WWW page.
The following mathematicians have helped me to compile this list. Joel Anderson, Jonathan Arazy,
Rajendra Bhatia, H P Boas, G D Bruechert, R B Burckel, Paul Chernoff, Norma Elias, George Gaspar,
Paul Halmos, Xun-Cheng Huang, M D Humphries, René Mata-Guarneros, Oisin McGuinness, David Minda,
Jeff Nichols, Billy Rhoades, Stephen Rowe, William Salkin, Glenn Schober, Karl Stromberg, Thad Tarpey,
Shelden Trimble, David Ullrich.
I would appreciate any further corrections or comments you wish to make.
Page Line
xii
23
1
7
5
-10
5
-3
From
Mondromy
supermum
in an nth √
√1 (−1 + i 3)
2
√
√1 (−1 − i 3)
2
its projection
one point
R
R
Let A ⊆ X ;
Let A ⊆ X ;
If K is a compact
are subsets
are disjoint sets
(Ω, p)
that f be
Any function
To
Monodromy
supremum
is an nth√
1
2 (−1 + i 3)
√
1
2 (−1 − i 3)
its stereographic projection
one end point
R
R
Let A be a non-empty subset of X;
Let A be a non-empty subset of X;
If K is a non-empty compact
are non-empty subsets
are non-empty disjoint sets
(Ω, ρ)
that g be
Any real-valued function
φ(s,t)+ψ(x,t)
s+it
φ(s,t)+iψ(s,t)
s+it
5
10
15
17
17
25
25
27
27
28
29
35
41
42
62
64
75
80
83
85
-3
11
14
18
20
-4
-2
13
-4
6
-4
23
-14
16
-9
-4
-2
-2
15
1
85
87
99
100
100
101
107
117
122
122
130
131
131
-14
10
-12
16
-6
-13
-8
-5
8
9
-13
17
22
138
140
141
141
141
4,6
-11
-10
-9
-9
141
-12
146
-6
Theorem II. 4.9 F
147
149
-2
-5
as k → ∞
be compact
denoted it by
C
(−1)n 2n
n
e2πint .
z≥R
By Lemma 5.1 . . . on C;
denoted by
R
(−1)n−1 2n
n
e2πint , 0 ≤ t ≤ 1.
|z| ≥ R
By Lemma 5.1 g is analytic on H
and by an analogue of Leibniz’s
rule (for example, see Exercise 2.2)
g is analytic on G;
Define g(z, w)
Define φ(z, w)
B(±1; 12 )
B̄(±1; 21 )
each z ∈ Ω
each z ∈ G
then
then
triangle
triangular
(II.3.6)
(II.3.7)
|z − a| > r1
|z − a| > R1
This can more easily be obtained by using the substitution x → 1/x .
decrease the spaces between cos2 and x and between sinh2 and y
decrease the spaces between sin2 and x and between sinh2 and y
Re f(z)
Re f (z)
D onto
D into
= ∂D.
= ∂D, and, from the preceding
material,φ(D) = D.
Circle
Circles
exp(|z)a
exp(|z|a
δ
limr→0
limr→∞
exp(−/r
exp(−/r)
∞
S
Kn
The ∞ is not clear
n=1
Theorem II.4.9, F
(also correct spacing)
as j → ∞
be a compact
150
-2
153
153
154
156
160
167
176
186
188
195
206
206
-11
-4
-1
17
18
-10
13
8
14
2
3
-12
is equicontinuous.
Ascoli-Arzela
|f (a) − f (z)| ≤
Show that
put M = |f (a)|
is analytic
theorem
situation).
(cos θ)2u−1 (sin θ)2v−1
δ>β>α
Theorem
k≤1
sequences of distinct points in G
209
-13
209
-5
209
-4
211
19
211
19
213
-13
213
-12
214 -9,-8
a free ideal
kn
k
Rn
RT f
f
P
x in G0 f (x)
G+
fs (z) = ft (z), z ∈ Ds ∩ Dt whenever
|s − t| < δ
215
22
But since
215
215
22
27
228
6
z in Dt ∩ Bt .
|s − t| < δ; so G = Dt ∩ Bt ∩ Ds ∩ Bs
contains γ(s) and, therefore,
is a non-empty open set.
for all z in B ∩ D
235
235
238
238
238
5
-1
13
-15
-12
239
239
241
4
24
16
247
248
248
249
253
254
260
261
10
20
-17
-2
-2
-9
-9
-10
(µ ◦ h−1 )
as in 6.3(c)
ψ(f (x)) ∈ F
let (V, φ) ∈ Φ such that
not constant.
6.3(c)
There F consists
continuation along γ
off [0, 1]
{(gt , At , )}
the component
2πi[1 − t)
then,
aδ
R2 −r 2
|Reit −reiθ |2
is equicontinuous at each
point of G.
Arzela-Ascoli
|f (a) − f (z)| =
If G is a region, show that
put M = |f (a)| + 1.
is an analytic
lemma
situation?)
(cos θ)2u−1 (sin θ)2v−1
δ>β>α>0
theorem
k≥1
sequences of distinct points in G
without limit points in G
a proper free ideal
kn + 1
k
Rn + 1
RT g
g
P
x in G0 , f (x)
G+
fs (z) = ft (z), whenever |s − t| < δ
and z belongs to the component
of Ds ∩ Dt that contains γ(s).
Let H be a connected subset of
Dt ∩ Bt which contains γ(s) and
γ(t). But since
z in H.
|s − t| < δ. Let G be a region such
that γ((t − δ, t + δ)) ⊆ G
⊆ Dt ∩ Bt ; in particular, γ(s) ∈ G.
for all z in the component of
B ∩ D that contains a
(µ ◦ h−1 )−1
as in 6.3(b)
φ(x) ∈ F
let (V, φ) ∈ Φ with a in V such that
not constant on any
component of φ(W ).
6.3(b)
Then F consists
continuation along γ with
each Dt a disk
of [0, 1]
{(gt , At )}
a component
2πi[(1 − t)
then
aδ
R2 −r 2
|R−reiθ |2
Eliminate the material from “If ρ < R then ...” to “for some
constant C.” on line -5. Substitute for this the following.
268
271
271
272
272
10
12
18
9
13
274
18
274
276
284
284
-8
-9
-6
-3
286
-5
287
287
288
288
14
-5
-12
-10
“If ρ < R, then, for m ≤ n, Harnack’s Inequality applied to the positive
harmonic function un − um implies there is a constant C depending
only on ρ and R such that 0 ≤ un (z) − um (z) ≤ C[un (a) − um (a)] for
|z − a| ≤ ρ.”
|v(z)| ≤ 1
v(z) ≤ 1
C∞ such that
C∞ − G such that
Theorem VIII.3.2(c)
Theorem VIII.2.2(c)
barrier at a.
barrier at 0.
(y − t/x)2
(y − t)2 /x2
∞
∞
S
S
{γm }
{γn }
n=1
n=1
to a harmonic function
f (znk ) → w
< α2 |z|µ+1
< 12 α|z|µ+1 .
∞
P
to a harmonic function h
f (znk ) → ω
≤ α2 |z|µ+1
< 12 α|z|µ+1 f or|z| > r3 .
∞
P
n−1
n=1
for some > 0.
for some > 0
f (0) = 1,
f (0) =
6 0
log 2n(r) ≤ log M (r)
(log 2)n(r) ≤ log M (2r)
Change the inequalities here from
log 2n(r)r−(p+1) ≤ log [M (r)]r−(p+1)
≤ r(λ+)−(p+1)
to
(log 2)n(r)r−(p+1) ≤ log [M (2r)]r−(p+1)
≤ r(λ+)−(p+1) 2λ+
289
-5
289
291
291
-1
-4
-1
295
296
297
300
301
301
301
308
317
-5
-2
6
4
25
-19
-17
-10
≤ log M (r). Since f has order λ,
1
log M (r) ≤ rλ+ 2 k −(p+1/λ+)
Use Exercise 2.9
with zeros {log 2, log 3, ...}
≤ log M (2r). Since f has order λ,
1
log M (2r) ≤ (2r)λ+ 2 k −(p+1)/(λ+)
Use Exercise 2.8
with zeros {log 2, log 3, ...} and no
other zeros.
= {a
= {z
a branch of g of
a branch g of
√
√
±2
∓2
n−1
n−1
Montel-Caratheodory
Montel-Carathéodory
value is possible)
value is possible
Corollary XI.3.8).
Corollary XI.3.8.
with one exception
with one possible exception
conformed
conformal
Add the following entry to the List of Symbols.
∂∞ G 129