Complex Analysis Qualifying Exam Questions
## Table of Contents
## Contents
Table of Contents 2\
1 Preface 9\
2 Topology and Functions of One Variable 9\
2.1 1 9\
2.2 2 9\
2.3 3 9\
2.4 4 10\
2.5 5 10\
2.6 6 10\
2.7 7 10\
2.8 8 10\
2.9 9 11\
2.10 1 11\
3 Several Variables 11\
3.1 1 11\
3.2 2 11\
3.3 3 12\
3.4 4 12\
3.5 5 12\
3.6 6 13\
3.7 7 13\
4 Several Variables: Extra Questions 13\
4.1 ? 13\
4.2 ? 13\
4.2.1 Tie’s Extra Questions: Fall 2009 14\
4.3 Spring 2020 HW 1 # 10 14\
5 Complex Arithmetic 14\
5.1 Spring 2020 HW 1 # 1 . 14\
5.2 Spring 2020 HW 1 # 2 . 15\
5.3 Spring 2020 HW 1 # 3 . 15\
5.4 Spring 2020 HW 1 # 4 . 15\
5.5 Spring 2020 HW 1 # 5 . 16\
5.6 Spring 2020 HW 1 # 6 . 16\
5.7 Spring 2020 HW 1 # 11 16\
6 Holomorphicity 17\
6.1 Spring 2020 HW 1 # 7 . 17\
6.2 Spring 2020 HW 1 # 8 .\
6.3 Spring 2020 HW 1 # 9 .\
6.3.1 Spring 20202 HW 2 # 2.6.10\
6.3.2 Spring 20202 HW 2 # 2.6.13\
6.3.3 Spring 20202 HW 2 # 2.6.14 18\
6.3.4 Spring 20202 HW 2 # 1 18\
6.3.5 Spring 20202 HW 2 # 2 . 18\
6.3.6 Spring 20202 HW 2 # 3 18\
6.3.7 Spring 20202 HW 2 # 5 19\
7 Integrals and Cauchy’s Theorem (8155d) 19\
7.1 2 19\
7.2 3 19\
7.3 4 19\
7.4 5 20\
7.5 6 20\
7.6 7 20\
7.7 8 20\
7.8 9 21\
7.9 10 21\
8 Extra 21\
8.1 ? 21\
8.1.1 Tie’s Extra Questions: Fall 2009 21\
8.1.2 Spring 2020 HW 2 # 2.6.5 22\
8.1.3 Spring 2020 HW 2 # 2.6.6\
8.1.4 Spring 2020 HW 2 # 2.6.7\
8.1.5 Spring 2020 HW 2 # 2.6.8\
8.1.6 Spring 2020 HW 2 # 2.6.9\
8.1.7 Spring 2020 HW 2 # 6\
8.1.8 Spring 2020 HW 2 # 7\
8.1.9 Spring 2020 HW 2 # 8\
8.1.10 Spring 2020 HW 2 # 9 (Cauchy’s Formula for Exterior Regions)\
8.1.11 Spring 2020 HW 2 # 10\
8.1.12 Spring 2020 HW 2 # 11 24\
8.1.13 Spring 2020 HW 2 # 12 24\
8.1.14 Spring 2020 HW 2 # 13 242525\
8.1.15 Entire and O of polynomial implies polynomial\
8.2 Uniform sequence implies uniform derivatives\
9 Liouville’s Theorem, Power Series (8155e) 25\
9.1 1 25\
9.2 2 26\
9.3 3 26\
9.4 4\
9.5 5\
9.6 6\
9.7 7\
9.8 8 272828\
9.9 9\
9.10 10\
10 Extra 28\
10.1 Tie’s Questions 28\
10.2 Tie’s Questions 28\
10.2.1 Tie’s Extra Questions: Fall 2009 29\
10.2.2 Tie’s Extra Questions: Fall 2009 29\
10.2.3 Tie’s Extra Questions: Fall 2009 29\
10.2.4 Spring 20202 HW 2 # 4 29\
10.2.5 Spring 20202 HW 3 # 1 29\
10.2.6 Spring 20202 HW 3 # 2 . 30\
10.2.7 Spring 20202 HW 3 # 3 30\
10.2.8 Spring 20202 HW 3 # 5 . 30\
11 Laurent Expansions and Singularities (8155f) 30\
11.1 1 30\
11.2 2 30\
11.3 3 31\
11.4 4 31\
11.5 5 31\
11.6 6 31\
11.7 7 31\
11.8 8 32\
12 Residues 32\
12.1 3233\
12.2\
12.3 33\
12.4 33\
12.5 33\
12.6 34\
12.7 34\
12.8 34\
12.9 34\
12.10 35\
12.11 35\
12.12 35\
12.13 35\
13 Extra Questions 36\
13.1 36\
13.2 36\
13.3\
13.4\
13.4.1 Tie’s Extra Questions: Fall 2009\
13.4.2 Tie’s Extra Questions: Fall 2009\
13.4.3 Tie’s Extra Questions: Fall 2009 38\
13.4.4 Spring 20202 HW 2 # 2.6.1 38\
13.4.5 Spring 20202 HW 2 # 2.6.2 39\
13.4.6 Spring 20202 HW 3 # 3.8.1 39\
13.4.7 Spring 20202 HW 3 # 3.8.2 40\
13.4.8 Spring 20202 HW 3 # 3.8.4 40\
13.4.9 Spring 20202 HW 3 # 3.8.5 40\
13.4.10 Spring 20202 HW 3 # 3.8.6 40\
13.4.11 Spring 20202 HW 3 # 3.8.7 40\
13.4.12 Spring 20202 HW 3 # 3.8.8 40\
13.4.13 Spring 20202 HW 3 # 3.8.9 41\
13.4.14 Spring 20202 HW 3 # 3.8.10 41\
13.4.15 Spring 20202 HW 3 # 6 42\
13.4.16 Spring 20202 HW 3 # 7 . 42\
13.4.17 Spring 20202 HW 3 # 10 42\
14 Conformal Maps (8155c) 42\
14.1 1 42\
14.2 2 42\
14.3 3 43\
14.4 4 43\
14.5 5 43\
14.6 6 43\
14.7 7 43\
14.8 8 43\
14.9 9 44\
14.1010 44\
14.1111 44\
14.1212 44\
14.1313 44\
14.13.1 Tie’s Extra Questions: Fall 2009 44\
15 Maps of the Disc 45\
15.1 Spring 2020 HW 1 # 5 . 45\
16 Rouche’s Theorem (8155h) 45\
16.1 1 45\
16.2 2 45\
16.3 3 46\
16.4 4 46\
16.5 5 46\
16.6 6 46\
16.7 7 47\
16.8 8 47\
16.9 9 47\
16.1010 47\
17 Extras 48\
17.1 ? 4848\
17.2 ?\
17.2.1 Tie’s Extra Questions: Fall 2009 48\
17.2.2 Spring 20202 HW 3 # 8 . 48\
17.2.3 Spring 20202 HW 3 # 11 48\
17.2.4 Spring 20202 HW 3 # 12 49\
17.2.5 Spring 20202 HW 3 # 13 49\
18 Schwarz Lemma and Reflection Principle (8155i) 49\
18.1 1 49\
18.1.1 a 49\
18.1.2 b 49\
18.1.3 c 49\
18.2 2 50\
18.3 3 50\
18.4 4 50\
18.5 5 50\
18.6 6 51\
18.7 7 51\
18.7.1 a 51\
18.7.2 b 51\
18.7.3 c 51\
18.8 8 51\
18.9 9 52\
18.1010 52\
18.10.1 Tie’s Extra Questions: Fall 2009 52\
18.10.2 Spring 20202 HW 2 # 2.6.15 52\
19 Riemann Mapping and Casorati-Weierstrass 53\
19.0.1 Spring 20202 HW 3 # 4 53\
19.1 10. 53\
19.1.1 Tie’s Extra Questions: Fall 2009 53\
19.1.2 Spring 20202 HW 3 # 3.8.14 53\
19.1.3 Spring 20202 HW 3 # 3.8.15 53\
19.1.4 Spring 20202 HW 3 # 3.8.17 54\
19.1.5 Spring 20202 HW 3 # 3.8.19 54\
19.1.6 Spring 20202 HW 3 # 9 . 55\
19.1.7 Spring 20202 HW 3 # 14 55\
20 Extra Questions from Jingzhi Tie 55\
20.1 16 55\
20.1.1 Tie’s Extra Questions: Fall 2009 55\
20.2 Fall 2009 . 56\
20.2.1 Tie’s Extra Questions: Fall 2009 56\
20.2.2 Tie’s Extra Questions: Fall 2009 5656\
20.2.3 Tie’s Extra Questions: Fall 2009\
20.2.4 Tie’s Extra Questions: Fall 2009 56\
20.3 Fall 2011 . 57\
20.3.1 Tie’s Extra Questions: Fall 2011\
20.3.2 Tie’s Extra Questions: Fall 2011\
20.3.3 Tie’s Extra Questions: Fall 2011\
20.3.4 Tie’s Extra Questions: Fall 2011 58\
20.3.5 Tie’s Extra Questions: Fall 2011 58\
20.3.6 Tie’s Extra Questions: Fall 2011 58\
20.3.7 Tie’s Extra Questions: Fall 2011 58\
20.3.8 Tie’s Extra Questions: Fall 2011 58\
20.3.9 Tie’s Extra Questions: Fall 2011 59\
20.3.10 Tie’s Extra Questions: Fall 2011 59\
20.3.11 Tie’s Extra Questions: Fall 2011 59\
20.3.12 Tie’s Extra Questions: Fall 2011 59\
20.3.13 Tie’s Extra Questions: Fall 2011 59\
20.4 Spring 2014 60\
20.4.1 Tie’s Extra Questions: Spring 2014 60\
20.4.2 Tie’s Extra Questions: Spring 2014 60\
20.4.3 Tie’s Extra Questions: Spring 2014 60\
20.4.4 Tie’s Extra Questions: Spring 2014 61\
20.4.5 Tie’s Extra Questions: Spring 2014 61\
20.4.6 Tie’s Extra Questions: Spring 2014 61\
20.4.7 Tie’s Extra Questions: Spring 2014 61\
20.4.8 Tie’s Extra Questions: Spring 2014 61\
20.4.9 Tie’s Extra Questions: Spring 2014 62\
20.4.10 Tie’s Extra Questions: Spring 2014 62\
20.5 Fall 2015 . . 62\
20.5.1 Tie’s Extra Questions: Fall 2015 62\
20.5.2 Tie’s Extra Questions: Fall 2015 62\
20.5.3 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 63\
20.5.4 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 63\
20.5.5 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 64\
20.5.6 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 64\
20.5.7 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 64\
20.5.8 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 64\
20.5.9 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 64\
20.5.10 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 65\
20.5.11 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 65\
20.5.12 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 65\
20.5.13 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 65\
20.5.14 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 65\
20.5.15 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 65\
20.5.16 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 66\
20.5.17 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 66\
20.5.18 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015 66\
20.6 Spring 2015 66\
20.6.1 Tie’s Extra Questions: Spring 2015 66\
20.6.2 Tie’s Extra Questions: Spring 2015 67\
20.6.3 Tie’s Extra Questions: Spring 2015 67\
20.6.4 Tie’s Extra Questions: Spring 2015 67\
20.6.5 Tie’s Extra Questions: Spring 2015\
20.6.6 Tie’s Extra Questions: Spring 2015\
20.6.7 Tie’s Extra Questions: Spring 2015\
20.6.8 Tie’s Extra Questions: Spring 2015 68\
20.6.9 Tie’s Extra Questions: Spring 2015 68\
20.6.10 Tie’s Extra Questions: Spring 2015 68\
20.6.11 Tie’s Extra Questions: Spring 2015 68\
20.6.12 Tie’s Extra Questions: Spring 2015 69\
20.6.13 Tie’s Extra Questions: Spring 2015 69\
20.6.14 Tie’s Extra Questions: Spring 2015 69\
20.6.15 Tie’s Extra Questions: Spring 2015 69\
20.6.16 Tie’s Extra Questions: Spring 2015 70\
20.6.17 Tie’s Extra Questions: Spring 2015 70\
20.6.18 Tie’s Extra Questions: Spring 2015 70\
20.6.19 Tie’s Extra Questions: Spring 2015 70\
20.6.20 Tie’s Extra Questions: Spring 2015 70\
20.6.21 Tie’s Extra Questions: Spring 2015 71\
20.6.22 Tie’s Extra Questions: Spring 2015 71\
20.6.23 Tie’s Extra Questions: Spring 2015 71\
20.6.24 Tie’s Extra Questions: Spring 2015 71\
20.6.25 Tie’s Extra Questions: Spring 2015 71\
20.6.26 Tie’s Extra Questions: Spring 2015 72\
20.6.27 Tie’s Extra Questions: Spring 2015 72\
20.6.28 Tie’s Extra Questions: Spring 2015\
20.6.29 Tie’s Extra Questions: Spring 2015\
20.6.30 Tie’s Extra Questions: Spring 2015\
20.6.31 Tie’s Extra Questions: Spring 2015\
20.6.32 Tie’s Extra Questions: Spring 2015 73\
20.6.33 Tie’s Extra Questions: Spring 2015 73\
20.7 Fall 2016 . 73\
20.7.1 Tie’s Extra Questions: Fall 2016 73\
20.7.2 Tie’s Extra Questions: Fall 2016 74\
20.7.3 Tie’s Extra Questions: Fall 2016 74\
20.7.4 Tie’s Extra Questions: Fall 2016 74\
20.7.5 Tie’s Extra Questions: Fall 2016 75\
20.7.6 Tie’s Extra Questions: Fall 2016 75\
20.7.7 Tie’s Extra Questions: Fall 2016 75\
20.7.8 Tie’s Extra Questions: Fall 2016 75\
21 Fall 2021 75\
21.1 Fall 2021 #1 75\
21.2 Fall 2021 #2 76\
21.3 Fall 2021 #3 76\
21.4 Fall 2021 #4 76\
21.5 Fall 2021 #5\
21.6 Fall 2021 #6\
21.7 Fall 2021 #7
## 1 Preface
I’d like to thank the following individuals for their contributions to this document:
• Edward Azoff, for supplying a problem sheet broken out by topic.
• Mentzelos Melistas, for explaining and documenting many solutions to these questions.
• Jingzhi Tie, for supplying many additional problems and solutions.
On notation: I sometimes use $f _ { x } : = { \frac { \partial f } { \partial x } }$ to denote partial derivatives, and ∂nz f as $f ^ { ( n ) } ( z )$
## 2 Topology and Functions of One Variable
## 2.1 1
Let $x _ { 0 } = a , x _ { 1 } = b$ , and set
$$
x _ { n } : = { \frac { x _ { n - 1 } + x _ { n - 2 } } { 2 } } \quad n \geq 2 .
$$
Show that $\{ x _ { n } \}$ is a Cauchy sequence and find its limit in terms of a and b.
2.2 2
Suppose $f : \mathbb { R } \to \mathbb { R }$ is continuous and $\operatorname* { l i m } _ { x \to \pm \infty } f ( x ) = 0$ . Prove that f is uniformly continuous.
2.3 3
Give an example of a function f : R → R that is everywhere differentiable but $f ^ { \prime }$ is not continuous at 0.
## 2.4 4
Suppose $\left\{ g _ { n } \right\}$ is a uniformly convergent sequence of functions from R to R and $f : \mathbb { R } $ R is uniformly continuous.
Prove that the sequence $\{ f \circ g _ { n } \}$ is uniformly convergent.
## 2.5 5
Let f be differentiable on [a, b]. Say that f is uniformly differentiable iff
$$
\forall \varepsilon > 0 , \exists \delta > 0 \ \mathrm { s u c h \ t h a t }
$$
$$
| x - y | < \delta \implies \left| { \frac { f ( x ) - f ( y ) } { x - y } } - f ^ { \prime } ( y ) \right| < \varepsilon .
$$
Prove that f is uniformly differentiable on $[ a , b ] \iff f ^ { \prime }$ is continuous on $[ a , b ]$
2.6 6
Suppose A, $B \subseteq \mathbb { R } ^ { n }$ are disjoint and compact.
Prove that there exist $a \in A , b \in B$ such that
$$
\| a - b \| = \operatorname* { i n f } \{ \| x - y \| \ \middle | \ | \ x \in A , y \in B \} .
$$
$$
2 . 7 ~ 7
$$
Suppose A, $B \subseteq \mathbb { R } ^ { n }$ are connected and not disjoint.
Prove that A ∪ B is also connected.
## 2.8 8
Suppose $\{ f _ { n } \} _ { n \in \mathbb { N } }$ is a sequence of continuous functions $f _ { n } : [ 0 , 1 ] \to \mathbb { R }$ such that
$$
f _ { n } ( x ) \geq f _ { n + 1 } ( x ) \geq 0 \quad \forall n \in \mathbb { N } , \forall x \in [ 0 , 1 ] .
$$
Prove that if $\{ f _ { n } \}$ converges pointwise to 0 on [0, 1] then it converges to 0 uniformly on [0, 1].
## 2.9 9
Show that if $E \subset [ 0 , 1 ]$ is uncountable, then there is some $t \in \mathbb { R }$ such that $E \cap ( - \infty , t )$ and $E \cap ( t , \infty )$ are also uncountable.
## 2.10 1
Suppose $f , g : [ 0 , 1 ] \to \mathbb { R }$ where f is Riemann integrable and for $x , y \in [ 0 , 1 ]$
$$
| g ( x ) - g ( y ) | \leq | f ( x ) - f ( y ) | .
$$
Prove that g is Riemann integrable.
## 3 Several Variables
## 3.1 1
Is the following function continuous, differentiable, continuously differentiable?
$$
\begin{array} { r l } & { f : \mathbb { R } ^ { 2 } \to \mathbb { R } } \\ & { f ( x , y ) = \left\{ \begin{array} { l l } { \displaystyle \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } } & { ( x , y ) \neq ( 0 , 0 ) } \\ { 0 } & { \mathrm { e l s e } . } \end{array} \right. } \end{array}
$$
## 3.2 2
a. Complete this definition: ${ } ^ { * } f : \mathbb { R } ^ { n } \mathbb { R } ^ { m }$ is real-differentiable a point $p \in \mathbb { R } ^ { n }$ iff there exists a linear transformation.
. . ”
b. Give an example of a function $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$ whose first-order partial derivatives exist everywhere but f is not differentiable at (0, 0).
c. Give an example of a function $f : \mathbb { R } ^ { 2 } \mathbb { R }$ which is real-differentiable everywhere but nowhere complex-differentiable.
## 3.3 3
Let $f : \mathbb { R } ^ { 2 } \to \mathbb { R } .$
a. Define in terms of linear transformations what it means for f to be differentiable at a point $( a , b ) \in \mathbb { R } ^ { 2 }$
b. State a version of the inverse function theorem in this setting.
c. Identify $\mathbb { R } ^ { 2 }$ with C and give a necessary and sufficient condition for a real-differentiable function at $( a , b )$ to be complex differentiable at the point $a + i b$ .
## 3.4 4
Let $f = u +$ iv be complex-differentiable with continuous partial derivatives at a point $z = r e ^ { i \theta }$ with $r \neq 0$ . Show that
$$
\frac { \partial u } { \partial r } = \frac { 1 } { r } \frac { \partial v } { \partial \theta } \qquad \frac { \partial v } { \partial r } = - \frac { 1 } { r } \frac { \partial u } { \partial \theta } .
$$
## 3.5 5
Let $P = ( 1 , 3 ) \in \mathbb { R } ^ { 2 }$ and define
$$
f ( s , t ) : = p s ^ { 3 } - 6 s t + t ^ { 2 } .
$$
a. State the conclusion of the implicit function theorem concerning $f ( s , t ) = 0$ when f is considered a function $\mathbb { R } ^ { 2 } \to \mathbb { R }$
b. State the above conclusion when f is considered a function $\mathbb { C } ^ { 2 } \to \mathbb { C } .$
c. Use the implicit function theorem for a function $\mathbb { R } \times \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { 2 }$ to prove (b).
Let $F : \mathbb { R } ^ { 2 } \mathbb { R }$ be continuously differentiable with $F ( 0 , 0 ) = 0$ and $\| \nabla F ( 0 , 0 ) \| < 1$
Prove that there is some real number $r > 0$ such that $| F ( x , y ) | < r$ whenever $\| ( x , y ) \| < r$
State the most general version of the implicit function theorem for real functions and outline how it can be proved using the inverse function theorem.
## 4 Several Variables: Extra Questions
## 4.1 ?
Let $f = u + i v$ be differentiable (i.e. $f ^ { \prime } ( z )$ exists) with continuous partial derivatives at a point $z = r e ^ { i \theta } , r \neq 0$
Show that
$$
\frac { \partial u } { \partial r } = \frac { 1 } { r } \frac { \partial v } { \partial \theta } , \quad \frac { \partial v } { \partial r } = - \frac { 1 } { r } \frac { \partial u } { \partial \theta } .
$$
4.2 ?
Give an example of a
Show that $f ( z ) = z ^ { 2 }$ is uniformly continuous in any open disk $| z | < R$ where $R > 0$ is fixed, but it is not uniformly continuous on C.
1. Show that the function $u = u ( x , y )$ given by
$$
u ( x , y ) = { \frac { e ^ { n y } - e ^ { - n y } } { 2 n ^ { 2 } } } \sin n x \quad { \mathrm { f o r ~ } } n \in \mathbf { N }
$$
is the solution on $D = \{ ( x , y ) ~ | x ^ { 2 } + y ^ { 2 } < 1 \}$ of the Cauchy problem for the Laplace equation
$$
\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0 , \quad u ( x , 0 ) = 0 , \quad \frac { \partial u } { \partial y } ( x , 0 ) = \frac { \sin n x } { n } .
$$
2. Show that there exist points $( x , y ) \in D$ such that $\operatorname* { i m } _ { n \to \infty } | u ( x , y ) | = \infty .$
## 4.2.1 Tie’s Extra Questions: Fall 2009
Let $f = u + i v$ be differentiable (i.e. f 0(z) exists) with continuous partial derivatives at a point $z = r e ^ { i \theta } , r \neq 0$ . Show that
$$
\frac { \partial u } { \partial r } = \frac { 1 } { r } \frac { \partial v } { \partial \theta } , \quad \frac { \partial v } { \partial r } = - \frac { 1 } { r } \frac { \partial u } { \partial \theta } .
$$
## 4.3 Spring 2020 HW 1 # 10
a. Show that in polar coordinates, the Cauchy-Riemann equations take the form
$$
\frac { \partial u } { \partial r } = \frac { 1 } { r } \frac { \partial v } { \partial \theta } \mathrm { ~ a n d ~ } \frac { \partial v } { \partial r } = - \frac { 1 } { r } \frac { \partial u } { \partial \theta } .
$$
b. Use (a) to show that the logarithm function, defined as
$$
\log z = \log r + i \theta { \mathrm { ~ w h e r e ~ } } z = r e ^ { i \theta } { \mathrm { ~ w i t h ~ } } - \pi < \theta < \pi .
$$
is holomorphic on the region $r > 0 , - \pi < \theta < \pi$
Also show that this function is not continuous in $r > 0$
## 5 Complex Arithmetic
## 5.1 Spring 2020 HW 1 # 1
Geometrically describe the following subsets of C:
$$
\begin{array} { l } { \mathrm { a . } \ \vert z - 1 \vert = 1 } \\ { \mathrm { b . } \ \vert z - 1 \vert = 2 \vert z - 2 \vert } \end{array}
$$
c. $1 / z = \bar { z }$
d. $\Re ( z ) = 3$
e. $\Im ( z ) = a$ with $a \in \mathbb { R } .$
f. $\Re ( z ) > a \mathrm { ~ w i t h ~ } a \in \mathbb { R } .$
g. $| z - 1 | < 2 | z - 2 |$
## 5.2 Spring 2020 HW 1 # 2
Prove the following inequality, and explain when equality holds:
$$
| z + w | \geq | | z | - | w | | .
$$
## 5.3 Spring 2020 HW 1 # 3
Prove that the following polynomial has its roots outside of the unit circle:
$$
p ( z ) = z ^ { 3 } + 2 z + 4 .
$$
Hint: What is the maximum value of the modulus of the first two terms $i f \left| z \right| \leq 1 \ell$
## 5.4 Spring 2020 HW 1 # 4
a. Prove that if $c > 0 .$
$$
| w _ { 1 } | = c | w _ { 2 } | \implies \Big | w _ { 1 } - c ^ { 2 } w _ { 2 } \Big | = c | w _ { 1 } - w _ { 2 } | .
$$
b. Prove that if $c > 0$ and $c \neq 1$ , with $z _ { 1 } \neq z _ { 2 }$ , then the following equation represents a circle:
$$
\left| { \frac { z - z _ { 1 } } { z - z _ { 2 } } } \right| = c .
$$
Find its center and radius.
Hint: use part (a)
## 5.5 Spring 2020 HW 1 # 5
a. Let $z , w \in \mathbb { C }$ with zw $\neq 1$ . Prove that
$$
\left| \frac { w - z } { 1 - \overline { { w } } z } \right| < 1 \mathrm { i f } \ | z | < 1 , \ | w | < 1
$$
with equality when $| z | = 1 \ \mathrm { o r } \ | w | = 1$
b. Prove that for a fixed $w \in \mathbb { D } ,$ , the mapping $F : z \mapsto { \frac { w - z } { 1 - { \overline { { w } } } z } }$ satisfies
• F maps D to itself and is holomorphic.
$F ( 0 ) = w$ and $F ( w ) = 0$
$| z | = 1$ implies $| F ( z ) | = 1$
## 5.6 Spring 2020 HW 1 # 6
Use nth roots of unity to show that
$$
2 ^ { n - 1 } \sin \left( { \frac { \pi } { n } } \right) \sin \left( { \frac { 2 \pi } { n } } \right) \cdot \cdot \cdot \sin \left( { \frac { ( n - 1 ) \pi } { n } } \right) = n .
$$
Hint:
$$
\begin{array} { c } { { 1 - \cos ( 2 \theta ) = 2 \sin ^ { 2 } ( \theta ) } } \\ { { 2 \sin ( 2 \theta ) = 2 \sin ( \theta ) \cos ( \theta ) . } } \end{array}
$$
## 5.7 Spring 2020 HW 1 # 11
Prove that the distinct complex numbers z1, z2, z3 are the vertices of an equilateral triangle if and only if
$$
z _ { 1 } ^ { 2 } + z _ { 2 } ^ { 2 } + z _ { 3 } ^ { 2 } = z _ { 1 } z _ { 2 } + z _ { 2 } z _ { 3 } + z _ { 3 } z _ { 1 } .
$$
## Holomorphicity
## 6.1 Spring 2020 HW 1 # 7
Prove that $f ( z ) = | z | ^ { 2 }$ has a derivative at z = 0 and nowhere else.
## 6.2 Spring 2020 HW 1 # 8
Let f (z) be analytic in a domain, and prove that f is constant if it satisfies any of the following conditions:
a. $| f ( z ) |$ is constant.
b. $\Re ( f ( z ) )$ is constant.
c. $\arg ( f ( z ) )$ is constant.
d. $\overline { { f ( z ) } }$ is analytic.
How do you generalize (a) and (b)?
## 6.3 Spring 2020 HW 1 # 9
Prove that $\operatorname { i f } z \mapsto f ( z )$ is analytic, then $z \mapsto { \overline { { f ( { \overline { { z } } } ) } } }$ is analytic.
## 6.3.1 Spring 20202 HW 2 # 2.6.10
Can every continuous function on D be uniformly approximated by polynomials in the variable z?
Hint: compare to Weierstrass for the real interval.
Solution omitted.
## 6.3.2 Spring 20202 HW 2 # 2.6.13
Suppose f is analytic, defined on all of C, and for each $z _ { 0 } \in \mathbb { C }$ there is at least one coefficient in the expansion $f ( z ) = \sum _ { n = 0 } ^ { \infty } c _ { n } ( z - z _ { 0 } ) ^ { n }$ is zero.
Prove that f is a polynomial.
Hint: use the fact that $c _ { n } n ! = f ^ { ( n ) } ( z _ { 0 } )$ and use a countability argument.
Solution omitted.
## 6.3.3 Spring 20202 HW 2 # 2.6.14
Suppose that f is holomorphic in an open set containing D except for a pole $z _ { 0 } \in \partial \mathbb { D }$ . Let $\sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n }$ be the power series expansion of f in D, and show that lim $\frac { a _ { n } } { a _ { n + 1 } } = z _ { 0 }$
Solution
## 6.3.4 Spring 20202 HW 2 # 1
Let $a _ { n } \neq 0$ and show that
$$
\operatorname* { l i m } _ { n \infty } { \frac { | a _ { n + 1 } | } { | a _ { n } | } } = L \implies \operatorname* { l i m } _ { n \infty } | a _ { n } | ^ { \frac { 1 } { n } } = L .
$$
In particular, this shows that when applicable, the ratio test can be used to calculate the radius of convergence of a power series.
## 6.3.5 Spring 20202 HW 2 # 2
Let f be a power series centered at the origin.
Prove that f has a power series expansion about any point in its disc of convergence.
## 6.3.6 Spring 20202 HW 2 # 3
Prove the following:
a. $\sum _ { n } n z ^ { n }$ does not converge at any point of $S ^ { 1 }$
b. $\sum _ { n } { \frac { z ^ { n } } { n ^ { 2 } } }$ converges at every point of $S ^ { 1 }$
c. $\sum _ { n } { \frac { z ^ { n } } { n } }$ converges at every point of $S ^ { 1 }$ except z = 1.
## 6.3.7 Spring 20202 HW 2 # 5
Assume f is continuous in the region $\{ x + i y \ | \ x \geq x _ { 0 } , \ 0 \leq y \leq b \}$ , and the following limit exists independent of y:
$$
\operatorname* { l i m } _ { x + \infty } f ( x + i y ) = A .
$$
Show that if $\gamma _ { x } : = \left\{ z = x + i t \ \Big \vert \ 0 \leq t \leq b \right\}$ , then
$$
\operatorname* { l i m } _ { x \to + \infty } \int _ { \gamma _ { x } } f ( z ) d z = i A b .
$$
## 7 Integrals and Cauchy’s Theorem (8155d)
## 7.1 2
State and prove Green’s Theorem for rectangles.
Use this to prove Cauchy’s Theorem for functions that are analytic in a rectangle.
## 7.2 3
Suppose $\{ f _ { n } \} _ { n \in \mathbb { N } }$ is a sequence of analytic functions on $\mathbb { D } : = \left\{ z \in \mathbb { C } \mid | z | < 1 \right\}$
Show that if $f _ { n } g$ for some $g : \mathbb { D } \to \mathbb { C }$ uniformly on every compact $K \subset \mathbb { D }$ , then g is analytic on D.
## 7.3 4
Suppose $\{ f _ { n } \} _ { n \in \mathbb { N } }$ is a sequence of entire functions where
$f _ { n } g$ pointwise for some $g : \mathbb { C } \to \mathbb { C }$
• On every line segment in $\mathbb { C } , f _ { n } g$ uniformly.
Show that
$g$ is entire, and
$f _ { n } g$ uniformly on every compact subset of C.
Prove that there is no sequence of polynomials that uniformly converge to $f ( z ) = { \frac { 1 } { z } } \ o n \ S ^ { 1 }$
Relevant concepts omitted.
Solution omitted.
## 7.5 6
Suppose that $f : \mathbb { R } \to \mathbb { R }$ is a continuous function that vanishes outside of some finite interval.
For each $z \in \mathbb { C } ,$ define
$$
g ( z ) = \int _ { - \infty } ^ { \infty } f ( t ) e ^ { - i z t } d t .
$$
Show that g is entire.
## 7.6 7
Suppose $f : \mathbb { C } \to \mathbb { C }$ is entire and
$$
| f ( z ) | \leq | z | ^ { \frac { 1 } { 2 } } \quad \mathrm { ~ w h e n ~ } | z | > 1 0 .
$$
Prove that f is constant.
## 7.7 8
Let $\gamma$ be a smooth curve joining two distinct points $a , b \in \mathbb { C }$
Prove that the function
$$
f ( z ) : = \int _ { \gamma } { \frac { g ( w ) } { w - z } } d w
$$
is analytic in $\mathbb { C } \setminus \gamma .$
## 7.8 9
Suppose that $f : \mathbb { C } \to \mathbb { C }$ is continuous everywhere and analytic on C \ R and prove that f is entire.
Something missing?
Solution omitted.
## 7.9 10
Prove Liouville’s theorem: suppose $f : \mathbb { C } \to \mathbb { C }$ is entire and bounded.
Use Cauchy’s formula to prove that $f ^ { \prime } \equiv 0$ and hence f is constant.
Solution omitted.
## 8 Extra
## 8.1 ?
Assume f is continuous in the region: $0 < | z - a | \leq R , 0 \leq \arg ( z - a )$ ≤ β0 $( 0 < \beta _ { 0 } \le 2 \pi )$ and the limit $\operatorname* { l i m } _ { z \to a } ( z - a ) f ( z ) = A $ exists.
Show that
$$
\operatorname * { l i m } _ { r 0 } \int _ { \gamma _ { r } } f ( z ) d z = i A \beta _ { 0 } \ ,
$$
where $\gamma _ { r } : = \lbrace z \mid z = a + r e ^ { i t } , 0 \leq t \leq \beta _ { 0 } \rbrace$
## 8.1.1 Tie’s Extra Questions: Fall 2009
Let f be a continuous function in the region
$$
D = \{ z \mid | z | > R , 0 \leq \arg z \leq \theta \} \quad \mathrm { w h e r e } \quad 1 \leq \theta \leq 2 \pi .
$$
If there exists k such that $\operatorname* { l i m } _ { z \to \infty } z f ( z ) = k$ for z in the region D. Show that
$$
\operatorname* { l i m } _ { R ^ { \prime } \to \infty } \int _ { L } f ( z ) d z = i \theta k ,
$$
where L is the part of the circle $| z | = R ^ { \prime }$ which lies in the region D.
## 8.1.2 Spring 2020 HW 2 # 2.6.5
Suppose $f \in C _ { \mathbb { C } } ^ { 1 } ( \Omega )$ and $T \subset \Omega$ is a triangle with $T ^ { \circ } \subset \Omega .$ . 1. Apply Green’s theorem to show that $\int _ { T } f ( z ) \ d z = 0 .$
2. Assume that f 0 is continuous and prove Goursat’s theorem.
Hint: Green’s theorem states
$$
\int _ { T } F d x + G d y = \int _ { T ^ { \circ } } \left( { \frac { \partial G } { \partial x } } - { \frac { \partial F } { \partial y } } \right) d x d y .
$$
## 8.1.3 Spring 2020 HW 2 # 2.6.6
Suppose that f is holomorphic on a punctured open set $\Omega \setminus \{ w _ { 0 } \}$ and let $T \subset \Omega$ be a triangle containing w0. Prove that if f is bounded near w0, then $\int _ { T } f ( z ) \ d z = 0$
## 8.1.4 Spring 2020 HW 2 # 2.6.7
Suppose $f : \mathbb { D } \to \mathbb { C }$ is holomorphic and let $d : = \operatorname* { s u p } _ { z , w \in \mathbb { D } } | f ( z ) - f ( w ) |$ be the diameter of the image of f. Show that $2 | f ^ { \prime } ( 0 ) | \leq d ,$ and that equality holds iff f is linear, so $f ( z ) = a _ { 1 } z + a _ { 2 }$
Hint:
$$
2 f ^ { \prime } ( 0 ) = { \frac { 1 } { 2 \pi i } } \int _ { | \xi | = r } { \frac { f ( \xi ) - f ( - \xi ) } { \xi ^ { 2 } } } \ d \xi
$$
whenever $0 < r < 1$
## 8.1.5 Spring 2020 HW 2 # 2.6.8
Suppose that f is holomorphic on the strip $S \ = \ \left\{ x + i y \ \big \vert \ x \in \mathbb { R } , \ - 1 < y < 1 \right\}$ with $| f ( z ) | \leq$ $A \left( 1 + | z | \right) ^ { \nu }$ for ν some fixed real number.
Show that for all $z \in S ,$ , for each integer $n \geq 0$ there exists an $A _ { n } \geq 0$ such that $\left| f ^ { ( n ) } ( x ) \right| \le A _ { n } ( 1 + | x | ) ^ { \nu }$ for all $x \in \mathbb { R }$
Hint: Use the Cauchy inequalities.
## 8.1.6 Spring 2020 HW 2 # 2.6.9
Let $\Omega \subset \mathbb { C }$ be open and bounded and $\varphi : \Omega \to \Omega$ holomorphic.
Prove that if there exists a point $z _ { 0 } \in \Omega$ such that $\varphi ( z _ { 0 } ) = z _ { 0 }$ and $\varphi ^ { \prime } ( z _ { 0 } ) = 1$ , then $\varphi$ is linear.
Hint: assume $z _ { 0 } = 0$ (explain why this can be done) and write $\varphi ( z ) = z + a _ { n } z ^ { n } + O ( z ^ { n + 1 } )$ near 0. Let $\varphi _ { k } = \varphi \circ \varphi \circ \cdots \circ \varphi$ and prove that $\varphi _ { k } ( z ) = z +$ $k a _ { n } z ^ { n } + O ( z ^ { n + 1 } )$ . Apply Cauchy’s inequalities and $l e t \ k \infty$ to conclude.
## 8.1.7 Spring 2020 HW 2 # 6
Show by example that there exists a function $f ( z )$ that is holomorphic on $\left\{ z \in \mathbb { C } \Big \vert 0 < | z | < 1 \right\}$ and for all $r < 1$ 1,
$$
\int _ { | z | = r } f ( z ) d z = 0 ,
$$
but $f$ is not holomorphic at $z = 0$
## 8.1.8 Spring 2020 HW 2 # 7
Let $f$ be analytic on a region R and suppose $f ^ { \prime } ( z _ { 0 } ) \neq 0$ for some $z _ { 0 } \in R$ . Show that if C is a circle of sufficiently small radius centered at $z _ { \mathrm { 0 } }$ , then
$$
\frac { 2 \pi i } { f ^ { \prime } \left( z _ { 0 } \right) } = \int _ { C } \frac { d z } { f ( z ) - f \left( z _ { 0 } \right) } .
$$
Hint: use the inverse function theorem.
## 8.1.9 Spring 2020 HW 2 # 8
Assume two functions $u , b : \mathbb { R } ^ { 2 } $ R have continuous partial derivatives at $( x _ { 0 } , y _ { 0 } )$ . Show that $f : = u +$ iv has derivative $f ^ { \prime } ( z _ { 0 } )$ at $z _ { 0 } = x _ { 0 } + i y _ { 0 }$ if and only if
$$
\operatorname * { l i m } _ { r 0 } \frac { 1 } { \pi r ^ { 2 } } \int _ { | z - z _ { 0 } | = r } f ( z ) d z = 0 .
$$
## 8.1.10 Spring 2020 HW 2 # 9 (Cauchy’s Formula for Exterior Regions)
Let $\gamma$ be a piecewise smooth simple closed curve with interior $\Omega _ { 1 }$ and exterior $\Omega _ { 2 }$ . Assume $f ^ { \prime }$ exists\
in an open set containing $\gamma$ and $\Omega _ { 2 }$ with lim $f ( z ) = A$ . Show that z→∞
$$
{ \frac { 1 } { 2 \pi i } } \int _ { \gamma } { \frac { f ( \xi ) } { \xi - z } } d \xi = { \left\{ \begin{array} { l l } { A , } & { { \mathrm { ~ i f ~ } } z \in \Omega _ { 1 } } \\ { - f ( z ) + A , } & { { \mathrm { ~ i f ~ } } z \in \Omega _ { 2 } } \end{array} \right. } .
$$
## 8.1.11 Spring 2020 HW 2 # 10
Let $f ( z )$ be bounded and analytic in C. Let $a \neq b$ be any fixed complex numbers.
Show that the following limit exists:
$$
\operatorname* { l i m } _ { R \to \infty } \int _ { | z | = R } { \frac { f ( z ) } { ( z - a ) ( z - b ) } } d z .
$$
Use this to show that $f ( z )$ must be constant.
## 8.1.12 Spring 2020 HW 2 # 11
Suppose $f ( z )$ is entire and
$$
\operatorname* { l i m } _ { z \to \infty } \frac { f ( z ) } { z } = 0 .
$$
Show that f (z) is a constant.
## 8.1.13 Spring 2020 HW 2 # 12
Let $f$ be analytic in a domain D and $\gamma$ be a closed curve in $D$ . For any $z _ { 0 } \in D$ not on γ, show that
$$
\int _ { \gamma } \frac { f ^ { \prime } ( z ) } { \left( z - z _ { 0 } \right) } d z = \int _ { \gamma } \frac { f ( z ) } { \left( z - z _ { 0 } \right) ^ { 2 } } d z .
$$
Give a generalization of this result.
## 8.1.14 Spring 2020 HW 2 # 13
Compute
$$
\int _ { | z | = 1 } \left( z + { \frac { 1 } { z } } \right) ^ { 2 n } { \frac { d z } { z } }
$$
and use it to show that
$$
\int _ { 0 } ^ { 2 \pi } \cos ^ { 2 n } ( \theta ) d \theta = 2 \pi \left( { \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot ( 2 n - 1 ) } { 2 \cdot 4 \cdot 6 \cdot \cdot \cdot ( 2 n ) } } \right) .
$$
## 8.1.15 Entire and O of polynomial implies polynomial $^ { + } _ { \nsucc }$
Problem 8.1.1 (?)\
Let $f ( z )$ be entire and assume that $| f ( z ) | \leq M | z | ^ { 2 }$ outside of some disk for some constant M . Show that f(z) is a polynomial in z of degree $\leq 2 .$
Solution omitted.
## 8.2 Uniform sequence implies uniform derivatives
Problem 8.2.1 (?)\
Let $a _ { n } ( z )$ be an analytic sequence in a domain D such that $\sum _ { n = 0 } ^ { \infty } \left| a _ { n } ( z ) \right|$ converges uniformly on bounded and closed sub-regions of D. Show that $\sum _ { n = 0 } ^ { \infty } \left| a _ { n } ^ { \prime } ( z ) \right|$ converges uniformly on bounded and closed sub-regions of D.
Solution omitted.
## 9 Liouville’s Theorem, Power Series (8155e)
## 9.1 1
Suppose f is analytic on a region Ω such that $\mathbb { D } \subseteq \Omega \subseteq \mathbb { C }$ and $f ( z ) = \sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n }$ is a power series with radius of convergence exactly 1.
a. $\downarrow$ Give an example of such an f that converges at every point of $S ^ { 1 }$
b. Give an example of such an f which is analytic at 1 but $\sum _ { n = 0 } ^ { \infty } a _ { n }$ diverges.
c. Prove that f can not be analytic at every point of $S ^ { 1 }$
## Missing part (c)
Solution omitted.
## 9.2 2
Suppose f is entire and has Taylor series $\sum a _ { n } z ^ { n }$ about 0.
a. 7 Express $a _ { n }$ as a contour integral along the circle $| z | = R$
b. Apply (a) to show that the above Taylor series converges uniformly on every bounded subset of C.
c. Determine those functions f for which the above Taylor series converges uniformly on all of C.
## 9.3 3
Suppose D is a domain and $f , g$ are analytic on D.
Prove that if $f g = 0$ on D, then either f ≡ 0 or g ≡ 0 on D.
## 9.4 4
Suppose f is analytic on $\mathbb { D } ^ { \circ }$ . Determine with proof which of the following are possible:
a. $f \left( { \frac { 1 } { n } } \right) = ( - 1 ) ^ { n }$ for each $n > 1$
b. $f \left( { \frac { 1 } { n } } \right) = e ^ { - n }$ for each even integer $n > 1$ while $f \left( { \frac { 1 } { n } } \right) = 0$ for each odd integer $n > 1$
c. $f \left( { \frac { 1 } { n ^ { 2 } } } \right) = { \frac { 1 } { n } }$ for each integer $n > 1$
d. $f \left( { \frac { 1 } { n } } \right) = { \frac { n - 2 } { n - 1 } }$ for each integer $n > 1$
## 9.5 5
Prove the Fundamental Theorem of Algebra (using complex analysis).
Solution omitted.
## 9.6 6
Find all entire functions that satisfy
$$
| f ( z ) | \geq | z | \quad \forall z \in \mathbb { C } .
$$
Prove this list is complete.
Solution omitted.
## 9.7 7
Suppose $\sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n }$ converges for some $z _ { 0 } \neq 0 .$
a. Prove that the series converges absolutely for each z with $| z | < | z | _ { 0 }$
b. Suppose $0 < r < | z _ { 0 } |$ and show that the series converges uniformly on $| z | \leq r$
## 9.8 8
Suppose f is entire and suppose that for some integer $n \geq 1$
$$
\operatorname* { l i m } _ { z \to \infty } \frac { f ( z ) } { z ^ { n } } = 0 .
$$
Prove that f is a polynomial of degree at most $n - 1$
## 9.9 9
Find all entire functions satisfying
$$
| f ( z ) | \leq | z | ^ { \frac { 1 } { 2 } } \quad { \mathrm { ~ f o r ~ } } | z | > 1 0 .
$$
## 9.10 10
Prove that the following series converges uniformly on the set $\left\{ z \ | \ \Im ( z ) < \ln 2 \right\}$
$$
\sum _ { n = 1 } ^ { \infty } { \frac { \sin ( n z ) } { 2 ^ { n } } } .
$$
## 10 Extra
## 10.1 Tie’s Questions
Let f(z) be entire and assume values of $f ( z )$ lie outside a bounded open set Ω. Show without using Picard’s theorems that f (z) is a constant.
Let $f ( z )$ be entire and assume values of $f ( z )$ lie outside a bounded open set Ω.
Show without using Picard’s theorems that f (z) is a constant.
## 10.2 Tie’s Questions
1. Assume $f ( z ) = \sum _ { n = 0 } ^ { \infty } c _ { n } z ^ { n }$ converges in $| z | < R .$
Show that for $r < R$
$$
{ \frac { 1 } { 2 \pi } } \int _ { 0 } ^ { 2 \pi } | f ( r e ^ { i \theta } ) | ^ { 2 } d \theta = \sum _ { n = 0 } ^ { \infty } | c _ { n } | ^ { 2 } r ^ { 2 n } .
$$
2. Deduce Liouville’s theorem from (1).
## 10.2.1 Tie’s Extra Questions: Fall 2009
Suppose f is entire and there exist A, $R > 0$ and natural number N such that
$$
| f ( z ) | \geq A | z | ^ { N } { \mathrm { ~ f o r ~ } } | z | \geq R .
$$
Show that
1. f is a polynomial and
2. the degree of f is at least $N$
## 10.2.2 Tie’s Extra Questions: Fall 2009
Let $f ( z )$ be entire and assume values of $f ( z )$ lie outside a bounded open set $\Omega .$ Show without using Picard’s theorems that $f ( z )$ is a constant.
## 10.2.3 Tie’s Extra Questions: Fall 2009
Let $f ( z )$ be entire and assume that $f ( z ) \leq M | z | ^ { 2 }$ outside some disk for some constant M. Show that f (z) is a polynomial in z of degree $\leq 2 .$
## 10.2.4 Spring 20202 HW 2 # 4
Without using Cauchy’s integral formula, show that ${ \mathrm { i f ~ } } | a | < r < | b |$ , then
$$
\int _ { \gamma } { \frac { d z } { ( z - \alpha ) ( z - \beta ) } } = { \frac { 2 \pi i } { \alpha - \beta } }
$$
where $\gamma$ denotes the circle centered at the origin of radius r with positive orientation.
Hint: take a Laurent expansion.
## 10.2.5 Spring 20202 HW 3 # 1
Prove that if f has two Laurent series expansions,
$$
f ( z ) = \sum c _ { n } ( z - a ) ^ { n } \quad { \mathrm { a n d } } \quad f ( z ) = \sum c _ { n } ^ { \prime } ( z - a ) ^ { n }
$$
then $c _ { n } = c _ { n } ^ { \prime }$
## 10.2.6 Spring 20202 HW 3 # 2
Find Laurent series expansions of
$$
\frac { 1 } { 1 - z ^ { 2 } } + \frac { 1 } { 3 - z }
$$
How many such expansions are there?
In what domains are each valid?
## 10.2.7 Spring 20202 HW 3 # 3
Let $P , Q$ be polynomials with no common zeros.
Assume a is a root of $Q .$ Find the principal part of $P / Q$ at $z = a$ in terms of $P$ and $Q$ if a is (1) a simple root, and (2) a double root.
## 10.2.8 Spring 20202 HW 3 # 5
Show that if f is entire and $\operatorname* { l i m } _ { z \to \infty } f ( z ) = \infty$ , then $f$ is a polynomial.
# 11 Laurent Expansions and Singularities(8155f)
$$
\mathbf { 1 1 . 1 \textbf { 1 } } \mathrm { \Omega }
$$
Find the Laurent expansion of
$$
f ( z ) = { \frac { z + 1 } { z ( z - 1 ) } }
$$
about $z = 0$ and $z = 1$ respectively.
Solution omitted.
$$
1 1 . 2 \ 2 \ \mathrm { ~ }
$$
Find the Laurent expansions about $z = 0$ of the following functions:
$$
e ^ { \frac { 1 } { z } }
$$
$$
\cos \left( { \frac { 1 } { z } } \right) .
$$
Solution omitted.
## 11.3 3
Find the Laurent expansion of
$$
f ( z ) = { \frac { z + 1 } { z ( z - 1 ) ^ { 2 } } }
$$
about z = 0 and z = 1 respectively.
Hint: recall that power series can be differentiated.
## 11.4 4
For the following functions, find the Laurent series about 0 and classify their singularities there:
$$
\sin ^ { 2 } ( z )
$$
$$
z \exp { \frac { z } { z ^ { 2 } } }
$$
$$
{ \frac { 1 } { z ( 4 - z ) } } .
$$
## 11.5 5
Find all entire functions with have poles at ∞.
## 11.6 6
Find all functions on the Riemann sphere that have a simple pole at z = 2 and a double pole at $z = \infty$ , but are analytic elsewhere.
## 11.7 7
Let f be entire, and discuss (with proofs and examples) the types of singularities f might have (removable, pole, or essential) at $z = \infty$ in the following cases:
1. f has at most finitely many zeros in $\mathbb { C } .$
2. f has infinitely many zeros in C.
## 11.8 8
Define
$$
f ( z ) = { \frac { \pi ^ { 2 } } { \sin ^ { 2 } \left( \pi z \right) } }
$$
$$
g ( z ) = \sum _ { n \in \mathbb { Z } } { \frac { 1 } { ( z - n ) ^ { 2 } } } .
$$
a. Show that f and g have the same singularities in C.
b. Show that f and g have the same singular parts at each of their singularities.
c. Show that $f , g$ each have period one and approach zero uniformly on $0 \leq x \leq 1$ as $| y | \infty$
d. Conclude that $f = g .$
Not finished.
Solution omitted.
## 12 Residues
12.1
Problem 12.1.1 (?)
Calculate
$$
\int _ { 0 } ^ { \infty } { \frac { 1 } { ( 1 + z ) ^ { 2 } ( z + 9 x ^ { 2 } ) } } d x .
$$
12.2
Problem 12.2.1 (?)
Let a > 0 and calculate
$$
\int _ { 0 } ^ { \infty } { \frac { x \sin ( x ) } { x ^ { 2 } + a ^ { 2 } } } d x .
$$
12.3
Problem 12.3.1 (?) Calculate
$$
\int _ { 0 } ^ { \infty } { \frac { \sqrt { x } } { ( x + 1 ) ^ { 2 } } } d x .
$$
12.4
Problem 12.4.1 (?) Calculate
$$
\int _ { 0 } ^ { \infty } { \frac { \cos ( x ) - \cos ( 4 x ) } { x ^ { 2 } } } d x .
$$
12.5
Problem 12.5.1 (?)
Let $a > 0$ and calculate
$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { 2 } } { ( x ^ { 2 } + a ^ { 2 } ) ^ { 2 } } } d x .
$$
12.6
Problem 12.6.1 (?) Calculate
Problem 12.8.1 (?) Calculate
Problem 12.7.1 (?) Calculate
Problem 12.9.1 (?) Calculate
$$
\int _ { 0 } ^ { \infty } { \frac { \sin ( x ) } { x } } d x .
$$
12.7
$$
\int _ { 0 } ^ { \infty } { \frac { \sin ( x ) } { x ( x ^ { 2 } + 1 ) } } d x .
$$
12.8
$$
\int _ { 0 } ^ { \infty } { \frac { \sqrt { x } } { 1 + x ^ { 2 } } } d x .
$$
12.9
$$
\int _ { - \infty } ^ { \infty } { \frac { 1 + x ^ { 2 } } { 1 + x ^ { 4 } } } d x .
$$
12.10
Problem 12.10.1 (?)
Let $a > 0$ and calculate
$$
\int _ { 0 } ^ { \infty } { \frac { \cos ( x ) } { ( x ^ { 2 } + a ^ { 2 } ) ^ { 2 } } } d x .
$$
## 12.11
Problem 12.11.1 (?)
Calculate
$$
\int _ { 0 } ^ { \infty } { \frac { \sin ^ { 3 } ( x ) } { x ^ { 3 } } } d x .
$$
## 12.12
Problem 12.12.1 (?)
Let $n \in \mathbb { Z } ^ { \geq 1 }$ and $0 < \theta < \pi$ and show that
$$
{ \frac { 1 } { 2 \pi i } } \int _ { | z | = 2 } { \frac { z ^ { n } } { 1 - 3 z \cos ( \theta ) + z ^ { 2 } } } d z = { \frac { \sin ( n \theta ) } { \sin ( \theta ) } } .
$$
## 12.13
Problem 12.13.1 (?)
Suppose $a > b > 0$ and calculate
$$
\int _ { 0 } ^ { 2 \pi } { \frac { 1 } { ( a + b \cos ( \theta ) ) ^ { 2 } } } d \theta .
$$
## 13 Extra Questions
## 13.1
Problem 13.1.1 (?)
Suppose that f is an analytic function in the region D which contains the point a. Let
$F ( z ) = z - a - q f ( z ) .$ , where q is a complex parameter.
1. Let $K \subset D$ be a circle with the center at point a and also we assume that $f ( z ) \neq 0$ for $z \in K$ . Prove that the function $F$ has one and only one zero $z = w$ on the closed disc $\overline { { K } }$ whose boundary is the circle K if
$$
| q | < \operatorname* { m i n } _ { z \in K } { \frac { | z - a | } { | f ( z ) | } } . .
$$
2. Let $G ( z )$ be an analytic function on the disk $\overline { { K } }$ . Apply the residue theorem to prove that
$$
\frac { G ( w ) } { F ^ { \prime } ( w ) } = \frac { 1 } { 2 \pi i } \int _ { K } \frac { G ( z ) } { F ( z ) } d z ,
$$
where w is the zero from (1).
## 13.2
Problem 13.2.1 (?)
Evaluate
$$
\int _ { 0 } ^ { \infty } { \frac { x \sin x } { x ^ { 2 } + a ^ { 2 } } } d x .
$$
13.3
Problem 13.3.1 (?)
Show that
$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { 1 + x ^ { n } } } d x = { \frac { \pi } { n \sin { \frac { a \pi } { n } } } }
$$
using complex analysis, $0 < a < n$ . Here n is a positive integer.
## 13.4
Problem 13.4.1 (?)
Show that
$$
\int _ { 0 } ^ { \infty } { \frac { \cos ( x ) } { x ^ { 2 } + b ^ { 2 } } } d x = { \frac { \pi e ^ { - b } } { 2 b } } .
$$
Solution omitted.
## 13.4.1 Tie’s Extra Questions: Fall 2009
Suppose that $f$ is an analytic function in the region D which contains the point a. Let
$F ( z ) = z - a - q f ( z )$ where q is a complex parameter.
(1) Let $K \subset D$ be a circle with the center at point a and also we assume that $f ( z ) \neq 0$ for $z \in K$ Prove that the function F has one and only one zero $z = w$ on the closed disc K whose boundary is the circle K ${ \mathrm { i f ~ } } | q | < \operatorname* { m i n } _ { z \in K } { \frac { | z - a | } { | f ( z ) | } }$
(2) Let $G ( z )$ be an analytic function on the disk $\overline { { K } }$ . Apply the residue theorem to prove that $\frac { G ( w ) } { F ^ { \prime } ( w ) } = \frac { 1 } { 2 \pi i } \int _ { K } \frac { G ( z ) } { F ( z ) } d z$ , where w is the zero from (1).
(3) $\mathrm { I f } ~ z \in K$ , prove that the function $\frac { 1 } { F ( z ) }$ can be represented as a convergent series with respect to q: ${ \frac { 1 } { F ( z ) } } = \sum _ { n = 0 } ^ { \infty } { \frac { ( q f ( z ) ) ^ { n } } { ( z - a ) ^ { n + 1 } } }$
## 13.4.2 Tie’s Extra Questions: Fall 2009
Evaluate
$$
\int _ { 0 } ^ { \infty } { \frac { x \sin x } { x ^ { 2 } + a ^ { 2 } } } d x .
$$
## 13.4.3 Tie’s Extra Questions: Fall 2009
Show that
$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { 1 + x ^ { n } } } d x = { \frac { \pi } { n \sin { \frac { a \pi } { n } } } }
$$
using complex analysis, $0 < a \le n$ . Here n is a positive integer.
## 13.4.4 Spring 20202 HW 2 # 2.6.1
Show that
$$
\int _ { 0 } ^ { \infty } \sin \left( x ^ { 2 } \right) d x = \int _ { 0 } ^ { \infty } \cos \left( x ^ { 2 } \right) d x = { \frac { \sqrt { 2 \pi } } { 4 } } .
$$
Hint: integrate $e ^ { - x ^ { 2 } }$ over the following contour, using the fact that $\int _ { - \infty } ^ { \infty } e ^ { - x ^ { 2 } } d x = { \sqrt { \pi } } .$
## 13.4.5 Spring 20202 HW 2 # 2.6.2
Show that
$$
\int _ { 0 } ^ { \infty } { \frac { \sin { x } } { x } } d x = { \frac { \pi } { 2 } } .
$$
fact that this integral equals $\frac { 1 } { 2 i } \int _ { - \infty } ^ { \infty } \frac { e ^ { i x } - 1 } { x } d x$ and integrate around an indented
## 13.4.6 Spring 20202 HW 3 # 3.8.1
Use the following formula to show that the complex zeros of sin(πz) are exactly the integers, and they are each of order 1:
$$
\sin \pi z = { \frac { e ^ { i \pi z } - e ^ { - i \pi z } } { 2 i } } .
$$
Calculate the residue of $\frac { 1 } { \sin ( \pi z ) }$ at $z = n \in \mathbb { Z }$
## 13.4.7 Spring 20202 HW 3 # 3.8.2
Evaluate the integral
$$
\int _ { \mathbb { R } } { \frac { d x } { 1 + x ^ { 4 } } } .
$$
What are the poles of $\frac { 1 } { 1 + z ^ { 4 } } ?$
## 13.4.8 Spring 20202 HW 3 # 3.8.4
Show that
$$
\int _ { - \infty } ^ { \infty } { \frac { x \sin x } { x ^ { 2 } + a ^ { 2 } } } d x = \pi e ^ { - a } , \quad { \mathrm { ~ f o r ~ a l l ~ } } a > 0 .
$$
## 13.4.9 Spring 20202 HW 3 # 3.8.5
Show that if $\xi \in \mathbb { R }$ , then
$$
\int _ { - \infty } ^ { \infty } \frac { e ^ { - 2 \pi i x \xi } } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } d x = \frac { \pi } { 2 } ( 1 + 2 \pi | \xi | ) e ^ { - 2 \pi | \xi | } .
$$
## 13.4.10 Spring 20202 HW 3 # 3.8.6
Show that
$$
\int _ { - \infty } ^ { \infty } { \frac { d x } { ( 1 + x ^ { 2 } ) ^ { n + 1 } } } = { \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot ( 2 n - 1 ) } { 2 \cdot 4 \cdot 6 \cdot \cdot \cdot ( 2 n ) } } \cdot \pi .
$$
## 13.4.11 Spring 20202 HW 3 # 3.8.7
Show that
$$
\int _ { 0 } ^ { 2 \pi } \frac { d \theta } { ( a + \cos \theta ) ^ { 2 } } = \frac { 2 \pi a } { ( a ^ { 2 } - 1 ) ^ { 3 / 2 } } , ~ \mathrm { w h e n e v e r } ~ a > 1 .
$$
## 13.4.12 Spring 20202 HW 3 # 3.8.8
Show that if $a , b \in \mathbb { R }$ with $a > | b |$ , then
$$
\int _ { 0 } ^ { 2 \pi } { \frac { d \theta } { a + b \cos \theta } } = { \frac { 2 \pi } { \sqrt { a ^ { 2 } - b ^ { 2 } } } } .
$$
## 13.4.13 Spring 20202 HW 3 # 3.8.9
Show that
$$
\int _ { 0 } ^ { 1 } \log ( \sin \pi x ) d x = - \log 2 .
$$
Figure 9. Contour in Exercise 9
## 13.4.14 Spring 20202 HW 3 # 3.8.10
Show that if $a > 0$ , then
$$
\int _ { 0 } ^ { \infty } { \frac { \log x } { x ^ { 2 } + a ^ { 2 } } } d x = { \frac { \pi } { 2 a } } \log a .
$$
## 13.4.15 Spring 20202 HW 3 # 6
a. Show (without using 3.8.9 in the S&S) that
$$
\int _ { 0 } ^ { 2 \pi } \log \left| 1 - e ^ { i \theta } \right| d \theta = 0
$$
b. Show that this identity is equivalent to S&S 3.8.9:
$$
\int _ { 0 } ^ { 1 } \log ( \sin ( \pi x ) ) \ d x = - \log 2 .
$$
## 13.4.16 Spring 20202 HW 3 # 7
Let $0 < a <$ 4 and evaluate
$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { \alpha - 1 } } { 1 + x ^ { 3 } } } d x
$$
## 13.4.17 Spring 20202 HW 3 # 10
For $a > 0$ , evaluate
$$
\int _ { 0 } ^ { \pi / 2 } { \frac { d \theta } { a + \sin ^ { 2 } \theta } }
$$
## 14 Conformal Maps (8155c)
Notation: D is the open unit disc, H is the open upper half-plane.
Find a conformal map from D to H.
## 14.2 2
Find a conformal map from the strip $\left\{ z \in \mathbb { C } \Big \vert 0 < \Im ( z ) < 1 \right\}$ to H.
## 14.3 3
Find a fractional linear transformation T which maps H to D, and explicitly describe the image of the first quadrant under T .
## 14.4 4
Find a conformal map from $\left\{ z \in \mathbb { C } \Big \vert | z - i | > 1 , \Re ( z ) > 0 \right\}$ to H.
## 14.5 5
Find a conformal map from $\{ z \in \mathbb { C } \ \middle \vert \ \vert z \vert < 1 , \middle \vert z - \frac { 1 } { 2 } > \frac { 1 } { 2 } \}$ to D.
14.6 6
Find a conformal map from $\left\{ \left| z - 1 \right| < 2 \right\} \cap \left\{ \left| z + 1 \right| < 2 \right\}$ to H.
## 14.7 7
Let Ω be the region inside the unit circle |z| = 1 and outside the circle $\left| z - { \frac { 1 } { 4 } } \right| = { \frac { 1 } { 4 } } .$
Find an injective conformal map from Ω onto some annulus $\{ r < | z | < 1 \}$ for constant r.
## 14.8 8
Let D be the region obtained by deleting the real interval [0, 1) from D; find a conformal map from D to D.
## 14.9 9
Find a conformal map from $\mathbb { C } \setminus \left\{ x \in \mathbb { R } \ \middle | \ x \leq 0 \right\}$ to D.
14.10 10
Find a conformal map from $\mathbb { C } \setminus \left\{ x \in \mathbb { R } \ \middle | \ x \geq 1 \right\}$ to D.
## 14.11 11
Find a bijective conformal map from G to H, where
$$
G : = \left\{ z \in \mathbb { C } \Big \vert | z - 1 | < \sqrt { 2 } , | z + 1 | < \sqrt { 2 } \right\} \backslash [ 0 , i ) .
$$
## 14.12 12
Prove that TFAE for a Möbius transformation T given by $T ( z ) = { \frac { a z + b } { c z + d } } \vdots$
a. T maps $\mathbb { R } \cup \{ \infty \}$ to itself.
b. It is possible to choose $a , b , c , d$ to be real numbers.
c. ${ \overline { { T ( z ) } } } = T ( { \bar { z } } )$ for every $z \in \mathbb { C P } ^ { 1 }$
d. There exist $\alpha \in \mathbb { R } , \beta \in \mathbb { C } \backslash$ R such that $T ( \alpha ) = \alpha \ { \mathrm { a n d } } \ T ( { \overline { { \beta } } } ) = { \overline { { T ( \beta ) } } }$
## 14.13 13
Find a conformal map from $D = \{ z : \ | z | < 1 , \ | z - 1 / 2 | > 1 / 2 \}$ to the unit disk $\Delta = \{ z : | z | < 1 \}$
## 14.13.1 Tie’s Extra Questions: Fall 2009
Find a conformal map from $D = \{ z : \ | z | < 1 , \ | z - 1 / 2 | > 1 / 2 \}$ to the unit disk $\Delta = \{ z : | z | < 1 \}$
## 15 Maps of the Disc
## 15.1 Spring 2020 HW 1 # 5
a. Let $z , w \in \mathbb { C }$ with $\bar { z } w \ne 1$ . Prove that
$$
\left| \frac { w - z } { 1 - \overline { { w } } z } \right| < 1 \mathrm { i f } \ | z | < 1 , \ | w | < 1
$$
with equality when $| z | = 1 \ \mathrm { o r } \ | w | = 1$
b. Prove that for a fixed $w \in \mathbb { D }$ , the mapping $F : z \mapsto { \frac { w - z } { 1 - { \overline { { w } } } z } }$ satisfies
• F maps D to itself and is holomorphic.
$F ( 0 ) = w$ and $F ( w ) = 0$
$| z | = 1$ implies $| F ( z ) | = 1$
## 16 Rouche’s Theorem (8155h)
## 16.1 1
Prove that for every $n \in \mathbb { Z } ^ { \geq 0 }$ the following polynomial has no roots in the open unit disc:
$$
f _ { n } ( x ) : = \sum _ { k = 0 } ^ { n } { \frac { z ^ { k } } { k ! } } .
$$
Hint: check $n = 1 , 2$ directly.
Solution omitted.
## 16.2 2
Assume that $| b | < 1$ and show that the following polynomial has exactly two roots (counting multiplicity) in $| z | < 1$ :
$$
f ( z ) : = z ^ { 3 } + 3 z ^ { 2 } + b z + b ^ { 2 } .
$$
Solution omitted.
## 16.3 3
Let $c \in \mathbb { C }$ with $| c | < { \frac { 1 } { 3 } } .$ Show that on the open set $\left\{ z \in \mathbb { C } \mid \Re ( z ) < 1 \right\}$ , the function $f ( z ) : = c e ^ { z }$ has exactly one fixed point.
## 16.4 4
How many roots does the following polynomial have in the open disc $| z | < 1 2$
$$
f ( z ) = z ^ { 7 } - 4 z ^ { 3 } - 1 .
$$
Solution omitted.
## 16.5 5
Let $n \in \mathbb { Z } ^ { \geq 0 }$ and show that the equation
$$
e ^ { z } = a z ^ { n }
$$
has n solutions in the open unit disc $\operatorname { i f } \left| a \right| > e ,$ and no solutions if $| a | < { \frac { 1 } { e } } .$
## 16.6 6
Let f be analytic in a domain D and fix $z _ { 0 } \in D$ with $w _ { 0 } : = f ( z _ { 0 } )$ . Suppose $z _ { \mathrm { 0 } }$ is a zero of $f ( z ) - w _ { 0 }$ with finite multiplicity m. Show that there exists $\delta > 0$ and $\varepsilon > 0$ such that for each w such that $0 < | w - w _ { 0 } | < \varepsilon ,$ the equation $f ( z ) - w = 0$ has exactly m distinct solutions inside the disc $| z - z _ { 0 } | < \delta .$
## 16.7 7
For $k = 1 , 2 , \cdots , n$ , suppose $| a _ { k } | < 1$ and
$$
f ( z ) : = \left( { \frac { z - a _ { 1 } } { 1 - { \bar { a } } _ { q } z } } \right) \left( { \frac { z - a _ { 2 } } { 1 - { \bar { a } } _ { 2 } z } } \right) \cdot \cdot \cdot \left( { \frac { z - a _ { n } } { 1 - { \bar { a } } _ { n } z } } \right) .
$$
Show that $f ( z ) = b$ has n solutions in $| z | < 1$
## 16.8 8
For each $n \in \mathbb { Z } ^ { \geq 1 }$ , let
$$
P _ { n } ( z ) = 1 + z + { \frac { 1 } { 2 ! } } z ^ { 2 } + \cdots + { \frac { 1 } { n ! } } z ^ { n } .
$$
Show that for sufficiently large $n ,$ the polynomial $P _ { n }$ has no zeros in $| z | < 1 0$ , while the polynomial $P _ { n } ( z ) - 1$ has precisely 3 zeros there.
## 16.9 9
Prove that
$$
\operatorname* { m a x } _ { | z | = 1 } \left| a _ { 0 } + a _ { 1 } z + \cdot \cdot \cdot + a _ { n - 1 } z ^ { n - 1 } + z ^ { n } \right| \ge 1 .
$$
Hint: the first part of the problem asks for a statement of Rouche’s theorem.
## 16.10 10
Use Rouche’s theorem to prove the Fundamental Theorem of Algebra.
## 17 Extras
Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra:
If
$$
P _ { n } ( z ) = a _ { 0 } + a _ { 1 } z + \cdots + a _ { n - 1 } z ^ { n - 1 } + a _ { n } z ^ { n } \quad ( a _ { n } \neq 0 )
$$
is a polynomial of degree $n ,$ then it has n zeros in C.
Suppose f is entire and there exist A, $R > 0$ and natural number N such that
$$
| f ( z ) | \geq A | z | ^ { N } { \mathrm { ~ f o r ~ } } | z | \geq R .
$$
Show that (i) f is a polynomial and (ii) the degree of f is at least N .
## 17.2.1 Tie’s Extra Questions: Fall 2009
Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra: If
$$
P _ { n } ( z ) = a _ { 0 } + a _ { 1 } z + \cdots + a _ { n - 1 } z ^ { n - 1 } + a _ { n } z ^ { n } \quad ( a _ { n } \neq 0 )
$$
is a polynomial of degree n, then it has n zeros in C.
## 17.2.2 Spring 20202 HW 3 # 8
Prove the fundamental theorem of Algebra using
a. Rouche’s Theorem.
b. The maximum modulus principle.
## 17.2.3 Spring 20202 HW 3 # 11
Find the number of roots of $p ( z ) = 4 z ^ { 4 } - 6 z + 3$ in $| z | < 1$ and $1 < | z | < 2$ respectively.
## 17.2.4 Spring 20202 HW 3 # 12
Prove that $z ^ { 4 } + 2 z ^ { 3 } - 2 z + 1 0$ has exactly one root in each open quadrant.
## 17.2.5 Spring 20202 HW 3 # 13
Prove that for a > 0, z tan z − a has only real roots.
# 18 Schwarz Lemma and Reflection Principle(8155i)
Suppose $f : \mathbb { D } \to \mathbb { D }$ is analytic and admits a continuous extension $\widetilde { f } : \overline { { \mathbb { D } } } \to \overline { { \mathbb { D } } }$ such that $| z | = 1 \implies$ $| f ( z ) | = 1$
$$
\mathbf { 1 8 . 1 . 1 \textbf { a } } ^ { \flat }
$$
Prove that f is a rational function.
## 18.1.2 b
Suppose that z = 0 is the unique zero of f. Show that
$$
\exists n \in \mathbb { N } , \lambda \in S ^ { 1 } \quad { \mathrm { s u c h t h a t } } \quad f ( z ) = \lambda z ^ { n } .
$$
$$
\mathbf { 1 8 . 1 . 3 ~ c ~ } ^ { \flat }
$$
Suppose that $a _ { 1 } , \cdots , a _ { n } \in \mathbb { D }$ are the zeros of f and prove that
$$
\exists \lambda \in S ^ { 1 } \quad { \mathrm { s u c h t h a t } } \quad f ( z ) = \lambda \prod _ { j = 1 } ^ { n } { \frac { z - a _ { j } } { 1 - { \overline { { a _ { j } } } } z } } .
$$
## 18.2 2
Let $\overline { { B } } ( a , r )$ denote the closed disc of radius r about $a \in \mathbb { C }$ . Let $f$ be holomorphic on an open set containing $\textstyle { \overline { { B } } } ( a , r )$ and let
$$
M : = \operatorname* { s u p } _ { z \in \overline { { B } } ( a , r ) } | f ( z ) | .
$$
Prove that
$$
z \in \overline { { B } } \left( a , \frac { r } { 2 } \right) , z \neq a , \qquad \frac { | f ( z ) - f ( a ) | } { | z - a | } \leq \frac { 2 M } { r } .
$$
## 18.3 3
Define
$$
G : = \left\{ z \in \mathbb { C } \Big \vert \ \Re ( z ) > 0 , | z - 1 | > 1 \right\} .
$$
Find all of the injective conformal maps $G \mathbb { D }$ . These may be expressed as compositions of maps, but explain why this list is complete.
## 18.4 4
Suppose $f : \mathbb { H } \cup \mathbb { R } \mathbb { C }$ satisfies the following:
$f ( i ) = i$
• f is continuous
• f is analytic on H
$f ( z ) \in \mathbb { R } \iff z \in \mathbb { R } .$
Show that f (H) is a dense subset of H.
## 18.5 5
Suppose $f : \mathbb { D } \to \mathbb { H }$ is analytic and satisfies $f ( 0 ) = 2$ . Find a sharp upper bound for $\left| f ^ { \prime } ( 0 ) \right|$ , and prove it is sharp by example.
## 18.6 6
Suppose $f : \mathbb { D } \to \mathbb { D }$ is analytic, has a single zero of order k at $z = 0 ,$ and satsifies $\operatorname* { l i m } _ { | z | \to 1 } | f ( z ) | = 1$ Give with proof a formula for $f ( z )$
## 18.7 7
## 18.7.1 a
State the standard Schwarz reflection principle involving reflection across the real axis.
## 18.7.2 b
Give a linear fractional transformation T mapping D to H. Let $g ( z ) = { \bar { z } } .$ , and show
$$
( T ^ { - 1 } \circ g \circ T ) ( z ) = 1 / { \bar { z } } .
$$
18.7.3 c
Suppose that f is holomorphic on D, continuous on ${ \overline { { \mathbb { D } } } } ,$ and real on $S ^ { 1 }$ . Show that f must be constant.
## 18.8 8
Suppose $f , g : \mathbb { D } \to \Omega$ are holomorphic with f injective and $f ( 0 ) = g ( 0 )$
Show that
$$
\forall 0 < r < 1 , \qquad g \left( \left\{ | z | < r \right\} \right) \subseteq f \left( \left\{ | z | < r \right\} \right) .
$$
## 18.9 9
Let $S : = \left\{ z \in \mathbb { D } \Big | \mathbb { S } ( z ) \geq 0 \right\}$ . Suppose $f : S \mathbb { C }$ is continuous on $S ,$ real on $S \cap \mathbb { R }$ , and holomorphic on $S ^ { \circ }$
Prove that f is the restriction of a holomorphic function on D.
## 18.10 10
Suppose $f : \mathbb { D } \to \mathbb { D }$ is analytic. Prove that
$$
\forall a \in \mathbb { D } , \qquad { \frac { | f ^ { \prime } ( a ) | } { 1 - | f ( a ) | ^ { 2 } } } \leq { \frac { 1 } { 1 - | a | ^ { 2 } } } .
$$
## 18.10.1 Tie’s Extra Questions: Fall 2009
Let g be analytic for $| z | \le 1$ and $| g ( z ) | < 1$ for $| z | = 1$
1. Show that g has a unique fixed point in $| z | < 1$
2. What happens if we replace $| g ( z ) | < 1$ with $| g ( z ) | \leq 1$ for $| z | = 1 2$ Give an example if (a) is not true or give an proof if (a) is still true.
3. What happens if we simply assume that f is analytic for $| z | < 1$ and $| f ( z ) | < 1$ for $| z | < 1 2$ Suppose that $f ( z ) \not \equiv z$ . Can f have more than one fixed point in $| z | < 1 2$
$$
H i n t : \ T h e \ m a p \ \psi _ { \alpha } ( z ) = \frac { \alpha - z } { 1 - \bar { \alpha } z } \ m a y \ b e \ u s e f u l .
$$
## 18.10.2 Spring 20202 HW 2 # 2.6.15
Suppose f is continuous and nonvanishing on D, and holomorphic in D. Prove that if $| z | = 1 \implies$ $| f ( z ) | = 1$ , then f is constant.
Hint: Extend f to all $o f \mathbb { C }$ by $f ( z ) = 1 / \overline { { f ( 1 / \bar { z } ) } }$ for any $| z | > 1$ and argue as in the Schwarz reflection principle.
# 19 Riemann Mapping andCasorati-Weierstrass
## 19.0.1 Spring 20202 HW 3 # 4
Let f be non-constant, analytic in $| z | > 0$ , where $f ( z _ { n } ) = 0$ for infinitely many points $z _ { n }$ with $\operatorname* { l i m } _ { n \to \infty } z _ { n } = 0$ J.
Show that $z = 0$ is an essential singularity for $f .$
Example: f(z) = sin(1/z).
## 19.1 10.
Let $f : \mathbb { C } \to \mathbb { C }$ be an injective analytic (also called univalent) function. Show that there exist complex numbers $a \neq 0$ and b such that $ f ( z ) = a z + b . $
## 19.1.1 Tie’s Extra Questions: Fall 2009
Let $f : \mathbb { C } \to \mathbb { C }$ be an injective analytic (also called univalent) function. Show that there exist complex numbers $a \neq 0$ and b such that $ f ( z ) = a z + b . $
## 19.1.2 Spring 20202 HW 3 # 3.8.14
Prove that all entire functions that are injective are of the form $ f ( z ) = a z + b$ with $a , b \in \mathbb { C }$ and $a \neq 0$
Hint: Apply the Casorati-Weierstrass theorem to $f ( 1 / z )$
## 19.1.3 Spring 20202 HW 3 # 3.8.15
Use the Cauchy inequalities or the maximum modulus principle to solve the following problems:
a. Prove that if f is an entire function that satisfies
$$
\operatorname* { s u p } _ { | z | = R } | f ( z ) | \leq A R ^ { k } + B
$$
for all $R > 0$ , some integer $k \geq 0$ , and some constants $A , B > 0$ , then $f$ is a polynomial of degree $\leq k$
b. Show that if f is holomorphic in the unit disc, is bounded, and converges uniformly to zero in the sector $\theta < \arg ( z ) < \varphi$ as $| z | \to 0$ , then $f \equiv 0$
c. Let $w _ { 1 } , \cdots w _ { n }$ be points on $S ^ { 1 } \subset \mathbb { C }$ . Prove that there exists a point $z \in S ^ { 1 }$ such that the product of the distances from z to the points $w _ { j }$ is at least 1.
Conclude that there exists a point $w \in S ^ { 1 }$ such that the product of the above distances is exactly 1.
d. Show that if the real part of an entire function is bounded, then f is constant.
## 19.1.4 Spring 20202 HW 3 # 3.8.17
Let f be non-constant and holomorphic in an open set containing the closed unit disc.
a. Show that if $| f ( z ) | = 1$ whenever $| z | = 1$ , then the image of f contains the unit disc.
Hint: Show that $f ( z ) ~ = ~ w _ { 0 }$ has a root for every
$w _ { 0 } \in \mathbb { D } .$ , for which it suffices to show that $f ( z ) = 0$
has a root. Conclude using the maximum modulus
principle.
b. If $| f ( z ) | \geq 1$ whenever $| z | = 1$ and there exists a $z _ { 0 } \in \mathbb { D }$ such that $| f ( z _ { 0 } ) | < 1$ , then the image of f contains the unit disc.
## 19.1.5 Spring 20202 HW 3 # 3.8.19
Prove that maximum principle for harmonic functions, i.e.
a. If u is a non-constant real-valued harmonic function in a region $\Omega ,$ then u can not attain a maximum or a minimum in Ω.
b. Suppose Ω is a region with compact closure $\overline { { \Omega } } .$ . If u is harmonic in Ω and continuous in ${ \overline { { \Omega } } } ,$ then
$$
\operatorname* { s u p } _ { z \in \Omega } | u ( z ) | \leq \operatorname* { s u p } _ { z \in \overline { { \Omega } } - \Omega } | u ( z ) | .
$$
Hint: to prove $( a ) ,$ assume u attains a local maximum at $z _ { 0 }$ Let f be holomorphic near $z _ { \mathrm { 0 } }$ with $\Re ( f ) = u ,$ and show that f is not an open map. Then (a) implies (b).
## 19.1.6 Spring 20202 HW 3 # 9
Let f be analytic in a region D and $\gamma \mathrm { ~ a ~ }$ rectifiable curve in D with interior in $D$ .
Prove that if f (z) is real for all $z \in \gamma$ , then f is constant.
## 19.1.7 Spring 20202 HW 3 # 14
Let $f$ be nonzero, analytic on a bounded region Ω and continuous on its closure $\overline { { \Omega } } .$
Show that if $| f ( z ) | \equiv M$ is constant for $z \in \partial \Omega$ , then $f ( z ) \equiv M e ^ { i \theta }$ for some real constant $\theta .$
## 20 Extra Questions from Jingzhi Tie
## 20.1 16
Let $f ( z )$ be analytic in an open set Ω except possibly at a point $z _ { \mathrm { 0 } }$ inside $\Omega .$
Show that if $f ( z )$ is bounded in near $z _ { \mathrm { 0 } }$ , then $\int _ { \Delta } f ( z ) d z = 0$ for all triangles $\Delta$ in Ω.
## 20.1.1 Tie’s Extra Questions: Fall 2009
For $s > 0$ , the gamma function is defined by $\Gamma ( s ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { s - 1 } d t .$
1. Show that the gamma function is analytic in the half-plane $\Re ( s ) > 0$ , and is still given there by the integral formula above.
2. Apply the formula in the previous question to show that
$$
\Gamma ( s ) \Gamma ( 1 - s ) = \frac { \pi } { \sin \pi s } .
$$
Hint: You may need $\Gamma ( 1 - s ) = t \int _ { 0 } ^ { \infty } e ^ { - v t } ( v t ) ^ { - s } d v$ for t > 0.
## 20.2 Fall 2009
## 20.2.1 Tie’s Extra Questions: Fall 2009
Let $f ( z )$ be analytic in an open set Ω except possibly at a point $z _ { \mathrm { 0 } }$ inside Ω. Show that if $f ( z )$ is bounded in near $z _ { \mathrm { 0 } }$ , then $\int _ { \Delta } f ( z ) d z = 0$ for all triangles $\Delta$ in Ω.
## 20.2.2 Tie’s Extra Questions: Fall 2009
Assume f is continuous in the region: $0 < | z - a | \leq R , \ 0 \leq \arg ( z - a ) \leq \beta _ { 0 } \ ( 0 < \beta _ { 0 } \leq 2 \pi )$ and the limit $\operatorname* { l i m } _ { z \to a } ( z - a ) f ( z ) = A $ exists. Show that
$$
\operatorname * { l i m } _ { r 0 } \int _ { \gamma _ { r } } f ( z ) d z = i A \beta _ { 0 } \ ,
$$
where $\gamma _ { r } : = \lbrace z \mid z = a + r e ^ { i t } , 0 \leq t \leq \beta _ { 0 } \rbrace$
## 20.2.3 Tie’s Extra Questions: Fall 2009
Show that $f ( z ) = z ^ { 2 }$ is uniformly continuous in any open disk $| z | < R$ where $R > 0$ is fixed, but it is not uniformly continuous on C.
## 20.2.4 Tie’s Extra Questions: Fall 2009
(1) Show that the function $u = u ( x , y )$ given by
$$
u ( x , y ) = { \frac { e ^ { n y } - e ^ { - n y } } { 2 n ^ { 2 } } } \sin n x \quad { \mathrm { f o r ~ } } n \in \mathbf { N }
$$
is the solution on $D = \{ ( x , y ) ~ | x ^ { 2 } + y ^ { 2 } < 1 \}$ of the Cauchy problem for the Laplace equation
$$
\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0 , \quad u ( x , 0 ) = 0 , \quad \frac { \partial u } { \partial y } ( x , 0 ) = \frac { \sin n x } { n } .
$$
(2) Show that there exist points $( x , y ) \in D$ such that lim n $_ { \stackrel { \mathrm { 1 S u p } } { \infty } } | u ( x , y ) | = \infty$
## 20.3 Fall 2011
## 20.3.1 Tie’s Extra Questions: Fall 2011
Let f be a continuous function in the region
$$
D = \{ z \ | | z | > R , 0 \leq \arg Z \leq \theta \} \quad \mathrm { w h e r e } \quad 0 \leq \theta \leq 2 \pi .
$$
If there exists k such that $\operatorname* { l i m } _ { z \to \infty } z f ( z ) = k$ for z in the region D. Show that
$$
\operatorname* { l i m } _ { R ^ { \prime } \to \infty } \int _ { L } f ( z ) d z = i \theta k ,
$$
where L is the part of the circle $| z | = R ^ { \prime }$ which lies in the region D.
## 20.3.2 Tie’s Extra Questions: Fall 2011
Suppose that f is an analytic function in the region D which contains the point a. Let
$$
F ( z ) = z - a - q f ( z ) , \quad { \mathrm { w h e r e } } \quad q { \mathrm { ~ i s ~ a ~ c o m p l e x ~ p a r a m e t e r . } }
$$
(1) Let $K \subset D$ be a circle with the center at point a and also we assume that $f ( z ) \neq 0$ for $z \in K$ Prove that the function F has one and only one zero $z = w$ on the closed disc $\overline { { K } }$ whose boundary is the circle K ${ \mathrm { i f ~ } } | q | < \operatorname* { m i n } _ { z \in K } { \frac { | z - a | } { | f ( z ) | } }$
(2) Let $G ( z )$ be an analytic function on the disk $\overline { { K } } .$ Apply the residue theorem to prove that $\frac { G ( w ) } { F ^ { \prime } ( w ) } = \frac { 1 } { 2 \pi i } \int _ { K } \frac { G ( z ) } { F ( z ) } d z$ , where w is the zero from (1).
(3) If $z \in K$ , prove that the function $\frac { 1 } { F ( z ) }$ can be represented as a convergent series with respect to q: ${ \frac { 1 } { F ( z ) } } = \sum _ { n = 0 } ^ { \infty } { \frac { ( q f ( z ) ) ^ { n } } { ( z - a ) ^ { n + 1 } } }$
## 20.3.3 Tie’s Extra Questions: Fall 2011
Evaluate $\int _ { 0 } ^ { \infty } { \frac { x \sin x } { x ^ { 2 } + a ^ { 2 } } } d x$
## 20.3.4 Tie’s Extra Questions: Fall 2011
Let $f = u + i v$ be differentiable $\mathrm { ( i . e . ~ } f ^ { \prime } ( z )$ exists) with continuous partial derivatives at a point $z = r e ^ { i \theta } , r \neq 0$ . Show that
$$
\frac { \partial u } { \partial r } = \frac { 1 } { r } \frac { \partial v } { \partial \theta } , \quad \frac { \partial v } { \partial r } = - \frac { 1 } { r } \frac { \partial u } { \partial \theta } .
$$
## 20.3.5 Tie’s Extra Questions: Fall 2011
Show that $\int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { 1 + x ^ { n } } } d x = { \frac { \pi } { n \sin { \frac { a \pi } { n } } } }$ using complex analysis, $0 \textless a \textless n$ . Here n is a positive integer.
## 20.3.6 Tie’s Extra Questions: Fall 2011
For $s > 0$ , the gamma function is defined by $\Gamma ( s ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { s - 1 } d t .$
1. Show that the gamma function is analytic in the half-plane $\Re ( s ) > 0$ , and is still given there by the integral formula above.
2. Apply the formula in the previous question to show that
$$
\Gamma ( s ) \Gamma ( 1 - s ) = \frac { \pi } { \sin \pi s } .
$$
Hint: You may need $\Gamma ( 1 - s ) = t \int _ { 0 } ^ { \infty } e ^ { - v t } ( v t ) ^ { - s } d v$ for t > 0.
## 20.3.7 Tie’s Extra Questions: Fall 2011
Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra: If
$$
P _ { n } ( z ) = a _ { 0 } + a _ { 1 } z + \cdots + a _ { n - 1 } z ^ { n - 1 } + a _ { n } z ^ { n } \quad ( a _ { n } \neq 0 )
$$
is a polynomial of degree n, then it has n zeros in C.
## 20.3.8 Tie’s Extra Questions: Fall 2011
Let g be analytic for $| z | \le 1$ and $| g ( z ) | < 1 \mathrm { f o r } | z | = 1$
• Show that g has a unique fixed point in $| z | < 1$
• What happens if we replace $| g ( z ) | < 1$ with $| g ( z ) | \leq 1$ for $| z | = 1 2$ Give an example if (a) is not true or give an proof if (a) is still true.
• What happens if we simply assume that f is analytic for $| z | < 1$ and $| f ( z ) | < 1$ for $| z | < 1 2$ Suppose that $f ( z ) \not \equiv z$ . Can f have more than one fixed point in $| z | < 1 2$
$$
H i n t : \ T h e \ m a p \ \psi _ { \alpha } ( z ) = \frac { \alpha - z } { 1 - \bar { \alpha } z } \ m a y \ b e \ u s e f u l .
$$
## 20.3.9 Tie’s Extra Questions: Fall 2011
Find a conformal map from $D = \{ z : \ | z | < 1 , \ | z - 1 / 2 | > 1 / 2 \}$ to the unit disk $\Delta = \{ z : | z | < 1 \}$
## 20.3.10 Tie’s Extra Questions: Fall 2011
Let $f ( z )$ be entire and assume that $f ( z ) \leq M | z | ^ { 2 }$ outside some disk for some constant M. Show that f (z) is a polynomial in z of degree $\leq 2$
## 20.3.11 Tie’s Extra Questions: Fall 2011
Let $f ( z )$ be analytic in an open set Ω except possibly at a point $z _ { \mathrm { 0 } }$ inside Ω. Show that if $f ( z )$ is bounded in near $z _ { \mathrm { 0 } }$ , then $\int _ { \Delta } f ( z ) d z = 0$ for all triangles $\Delta$ in Ω.
## 20.3.12 Tie’s Extra Questions: Fall 2011
Assume f is continuous in the region: $0 < | z - a | \leq R , \ 0 \leq \arg ( z - a ) \leq \beta _ { 0 } \ ( 0 < \beta _ { 0 } \leq 2 \pi )$ and the limit $\operatorname* { l i m } _ { z \to a } ( z - a ) f ( z ) = A $ exists. Show that
$$
\operatorname * { l i m } _ { r 0 } \int _ { \gamma _ { r } } f ( z ) d z = i A \beta _ { 0 } \ ,
$$
where $\gamma _ { r } : = \lbrace z \mid z = a + r e ^ { i t } , 0 \leq t \leq \beta _ { 0 } \rbrace$
## 20.3.13 Tie’s Extra Questions: Fall 2011
Show that $f ( z ) = z ^ { 2 }$ is uniformly continuous in any open disk $| z | < R$ where $R > 0$ is fixed, but it is not uniformly continuous on C.
(1) Show that the function $u = u ( x , y )$ given by
$$
u ( x , y ) = { \frac { e ^ { n y } - e ^ { - n y } } { 2 n ^ { 2 } } } \sin n x \quad { \mathrm { f o r ~ } } n \in \mathbf { N }
$$
is the solution on $D = \{ ( x , y ) ~ | x ^ { 2 } + y ^ { 2 } < 1 \}$ of the Cauchy problem for the Laplace equation
$$
\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0 , \quad u ( x , 0 ) = 0 , \quad \frac { \partial u } { \partial y } ( x , 0 ) = \frac { \sin n x } { n } .
$$
(2) Show that there exist points $( x , y ) \in D$ such that $\operatorname* { l i m } _ { n \to \infty } \operatorname* { s u p } _ { } | u ( x , y ) | = \infty .$
## 20.4 Spring 2014
## 20.4.1 Tie’s Extra Questions: Spring 2014
The question provides some insight into Cauchy’s theorem. Solve the problem without using the Cauchy theorem.
1. Evaluate the integral $\int _ { \gamma } z ^ { n } d z$ for all integers n. Here $\gamma$ is any circle centered at the origin with the positive (counterclockwise) orientation.
2. Same question as (a), but with $\gamma$ any circle not containing the origin.
3. Show that if $\vert a \vert < r < \vert b \vert$ , then $\int _ { \gamma } { \frac { d z } { ( z - a ) ( z - b ) } } d z = { \frac { 2 \pi i } { a - b } }$ . Here $\gamma$ denotes the circle centered at the origin, of radius r, with the positive orientation.
## 20.4.2 Tie’s Extra Questions: Spring 2014
Evaluate $\int _ { 0 } ^ { \infty } { \frac { x \sin x } { x ^ { 2 } + a ^ { 2 } } } d x .$
## 20.4.3 Tie’s Extra Questions: Spring 2014
Let $f = u +$ iv be differentiable (i.e. $f ^ { \prime } ( z )$ exists) with continuous partial derivatives at a point $z = r e ^ { i \theta } , r \neq 0$ . Show that
$$
\frac { \partial u } { \partial r } = \frac { 1 } { r } \frac { \partial v } { \partial \theta } , \quad \frac { \partial v } { \partial r } = - \frac { 1 } { r } \frac { \partial u } { \partial \theta } .
$$
## 20.4.4 Tie’s Extra Questions: Spring 2014
Show that $\int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { 1 + x ^ { n } } } d x = { \frac { \pi } { n \sin { \frac { a \pi } { n } } } }$ using complex analysis, $0 \textless a \textless n$ . Here n is a positive integer.
## 20.4.5 Tie’s Extra Questions: Spring 2014
For $s > 0$ , the gamma function is defined by $\Gamma ( s ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { s - 1 } d t .$
• Show that the gamma function is analytic in the half-plane $\Re ( s ) > 0$ , and is still given there by the integral formula above.
• Apply the formula in the previous question to show that
$$
\Gamma ( s ) \Gamma ( 1 - s ) = \frac { \pi } { \sin \pi s } .
$$
Hint: You may need $\Gamma ( 1 - s ) = t \int _ { 0 } ^ { \infty } e ^ { - v t } ( v t ) ^ { - s } d v$ for t > 0.
## 20.4.6 Tie’s Extra Questions: Spring 2014
Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra: If
$$
P _ { n } ( z ) = a _ { 0 } + a _ { 1 } z + \cdots + a _ { n - 1 } z ^ { n - 1 } + a _ { n } z ^ { n } \quad ( a _ { n } \neq 0 )
$$
is a polynomial of degree n, then it has n zeros in C.
## 20.4.7 Tie’s Extra Questions: Spring 2014
Suppose f is entire and there exist $A , R > 0$ and natural number N such that
$$
| f ( z ) | \geq A | z | ^ { N } { \mathrm { ~ f o r ~ } } | z | \geq R .
$$
Show that (i) f is a polynomial and (ii) the degree of f is at least N .
## 20.4.8 Tie’s Extra Questions: Spring 2014
Let $f : \mathbb { C } \to \mathbb { C }$ be an injective analytic (also called univalent) function.
Show that there exist complex numbers $a \neq 0$ and b such that $ f ( z ) = a z + b$
## 20.4.9 Tie’s Extra Questions: Spring 2014
Let g be analytic for $| z | \le 1$ and $| g ( z ) | < 1 \mathrm { f o r } | z | = 1$
• Show that g has a unique fixed point in $| z | < 1$
• What happens if we replace $| g ( z ) | < 1$ with $| g ( z ) | \leq 1$ for $| z | = 1 2$ Give an example if (a) is not true or give an proof if (a) is still true.
• What happens if we simply assume that f is analytic for $| z | < 1$ and $| f ( z ) | < 1$ for $| z | < 1 2$ Suppose that $f ( z ) \not \equiv z$ . Can f have more than one fixed point in $| z | < 1 2$
$$
H i n t : \ T h e \ m a p \ \psi _ { \alpha } ( z ) = \frac { \alpha - z } { 1 - \bar { \alpha } z } \ m a y \ b e \ u s e f u l .
$$
## 20.4.10 Tie’s Extra Questions: Spring 2014
Find a conformal map from $D = \{ z : \ | z | < 1 , \ | z - 1 / 2 | > 1 / 2 \}$ to the unit disk $\Delta = \{ z : | z | < 1 \}$
## 20.5 Fall 2015
## 20.5.1 Tie’s Extra Questions: Fall 2015
Let $a _ { n } \neq 0$ and assume that $\operatorname* { l i m } _ { n \to \infty } { \frac { | a _ { n + 1 } | } { | a _ { n } | } } = L$ . Show that lim ${ \sqrt [ n ] { | a _ { n } | } } = L$ . In particular, this shows n→∞\
that when applicable, the ratio test can be used to calculate the radius of convergence of a power\
series.
## 20.5.2 Tie’s Extra Questions: Fall 2015
(a) Let z, w be complex numbers, such that zw $\neq 1$ . Prove that
$$
\left| { \frac { w - z } { 1 - \overline { { w } } z } } \right| < 1 \ \mathrm { ~ i f ~ } | z | < 1 \ \mathrm { a n d ~ } | w | < 1 ,
$$
and also that
$$
\left| { \frac { w - z } { 1 - \overline { { w } } z } } \right| = 1 \ \mathrm { ~ i f ~ } | z | = 1 \mathrm { ~ o r ~ } | w | = 1 .
$$
(b) Prove that for fixed w in the unit disk D, the mapping
$$
F : z \mapsto { \frac { w - z } { 1 - w z } }
$$
satisfies the following conditions:
(c) F maps D to itself and is holomorphic.
(ii) F interchanges 0 and w, namely, $F ( 0 ) = w$ and $F ( w ) = 0$
(iii) $| F ( z ) | = 1 { \mathrm { ~ i f ~ } } | z | = 1$
(iv) $F : \mathbb { D } \mapsto \mathbb { D }$ is bijective.
Hint: Calculate F ◦ F .
## 20.5.3 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Use n-th roots of unity (i.e. solutions of $z ^ { n } - 1 = 0 )$ to show that
$$
2 ^ { n - 1 } \sin { \frac { \pi } { n } } \sin { \frac { 2 \pi } { n } } \cdot \cdot \cdot \sin { \frac { ( n - 1 ) \pi } { n } } = n .
$$
$$
H i n t \colon 1 - \cos 2 \theta = 2 \sin ^ { 2 } \theta , \sin 2 \theta = 2 \sin \theta \cos \theta .
$$
(a) Show that in polar coordinates, the Cauchy-Riemann equations take the form
$$
\frac { \partial u } { \partial r } = \frac { 1 } { r } \frac { \partial v } { \partial \theta } \mathrm { a n d } \frac { \partial v } { \partial r } = - \frac { 1 } { r } \frac { \partial u } { \partial \theta }
$$
(b) Use these equations to show that the logarithm function defined by
$$
\log z = \log r + i \theta \mathrm { \ w h e r e \ } z = r e ^ { i \theta } \mathrm { \ w i t h \ } - \pi < \theta < \pi
$$
is a holomorphic function in the region $r > 0 , - \pi < \theta < \pi$ . Also show that log z defined above is not continuous in $r > 0 .$ .
## 20.5.4 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Assume f is continuous in the region: $x \geq x _ { 0 } , 0 \leq y \leq b$ and the limit
$$
\operatorname* { l i m } _ { x + \infty } f ( x + i y ) = A
$$
exists uniformly with respect to y (independent of $y )$ . Show that
$$
\operatorname* { l i m } _ { x + \infty } \int _ { \gamma _ { x } } f ( z ) d z = i A b ,
$$
where $\gamma _ { x } : = \{ z \mid z = x + i t , 0 \leq t \leq b \}$
## 20.5.5 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
(Cauchy’s formula for “exterior” region) Let γ be piecewise smooth simple closed curve with interior $\Omega _ { 1 }$ and exterior $\Omega _ { 2 }$ . Assume $f ^ { \prime } ( z )$ exists in an open set containing $\gamma$ and $\Omega _ { 2 }$ and $\operatorname* { l i m } _ { z \to \infty } f ( z ) = A$ Show that
$$
{ \frac { 1 } { 2 \pi i } } \int _ { \gamma } { \frac { f ( \xi ) } { \xi - z } } d \xi = { \left\{ \begin{array} { l l } { A , } & { { \mathrm { i f ~ } } z \in \Omega _ { 1 } , } \\ { - f ( z ) + A , } & { { \mathrm { i f ~ } } z \in \Omega _ { 2 } } \end{array} \right. }
$$
## 20.5.6 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Let $f ( z )$ be bounded and analytic in C. Let $a \neq b$ be any fixed complex numbers.
Show that the following limit exists
$$
\operatorname* { l i m } _ { R \to \infty } \int _ { | z | = R } { \frac { f ( z ) } { ( z - a ) ( z - b ) } } d z .
$$
Use this to show that $f ( z )$ must be a constant (Liouville’s theorem).
## 20.5.7 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Prove by justifying all steps that for all $\xi \in \mathbb { C }$ we have $e ^ { - \pi \xi ^ { 2 } } = \int _ { - \infty } ^ { \infty } e ^ { - \pi x ^ { 2 } } e ^ { 2 \pi i x \xi } d x$
Hint: You may use that fact in Example 1 on $p . \ 4 2$ of the textbook without proof, i.e., you may assume the above is true for real values of ξ.
## 20.5.8 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Suppose that f is holomorphic in an open set containing the closed unit disc, except for a pole at $z _ { \mathrm { 0 } }$ on the unit circle.
Let denote the the power series in the open disc.
Show that $( 1 ) \ c _ { n } \neq 0$ for all large enough $n ' \mathrm { s } ,$ and (2) $\operatorname* { l i m } _ { n \to \infty } \frac { c _ { n } } { c _ { n + 1 } } = \bar { z } _ { 0 }$
## 20.5.9 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Let $f ( z )$ be a non-constant analytic function in $| z | > 0$ such that $f ( z _ { n } ) = 0$ for infinite many points\
$z _ { n }$ with lim $z _ { n } = 0$ . Show that $z = 0$ is an essential singularity for $f ( z )$ . (An example of such a n→∞\
function is $\begin{array} { r } { f ( z ) = \sin ( 1 / z ) . } \end{array}$ )
## 20.5.10 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Let f be entire and suppose that $\operatorname* { l i m } _ { z \to \infty } f ( z ) = \infty$ . Show that f is a polynomial.
## 20.5.11 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Expand the following functions into Laurent series in the indicated regions:
$$
f ( z ) = \frac { z ^ { 2 } - 1 } { ( z + 2 ) ( z + 3 ) } , 2 < | z | < 3 , 3 < | z | < + \infty .
$$
$$
{ \mathrm { ( b ) } } \ f ( z ) = \sin { \frac { z } { 1 - z } } , \ 0 < | z - 1 | < + \infty
$$
## 20.5.12 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Assume $f ( z )$ is analytic in region D and Γ is a rectifiable curve in D with interior in D. Prove that if $f ( z )$ is real for all $z \in \Gamma$ , then $f ( z )$ is a constant.
## 20.5.13 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Find the number of roots of $z ^ { 4 } - 6 z + 3 = 0 \ \mathrm { i n } \ | z | < 1$ and $1 < | z | < 2$ respectively.
## 20.5.14 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Prove that $z ^ { 4 } + 2 z ^ { 3 } - 2 z + 1 0 = 0$ has exactly one root in each open quadrant.
## 20.5.15 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
(1) Let $f ( z ) \in H ( \mathbb { D } ) , \operatorname { R e } ( f ( z ) ) > 0 , f ( 0 ) = a > 0$ . Show that
$$
| \frac { f ( z ) - a } { f ( z ) + a } | \leq | z | , | f ^ { \prime } ( 0 ) | \leq 2 a .
$$
(2) Show that the above is still true if $\operatorname { R e } ( f ( z ) ) > 0$ is replaced with $\mathrm { R e } ( f ( z ) ) \geq 0$
## 20.5.16 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Assume $f ( z )$ is analytic in D and $f ( 0 ) = 0$ and is not a rotation $( { \mathrm { i . e . ~ } } f ( z ) \neq e ^ { i \theta } z )$ . Show that $\sum ^ { \infty } f ^ { n } ( z )$ converges uniformly to an analytic function on compact subsets of D, where $f ^ { n + 1 } ( z ) =$ n=1\
${ \ddot { f } } ( { \bar { f } } ^ { n } ( z ) )$
## 20.5.17 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Let $f ( z ) = \sum _ { n = 0 } ^ { \infty } c _ { n } z ^ { n }$ be analytic and one-to-one in $| z | < 1$ . For $0 < r < 1$ , let $D _ { r }$ be the disk $| z | < r$ Show that the area of $f ( D _ { r } )$ is finite and is given by
$$
S = \pi \sum _ { n = 1 } ^ { \infty } n | c _ { n } | ^ { 2 } r ^ { 2 n } .
$$
(Note that in general the area of $f ( D _ { 1 } )$ is infinite.)
## 20.5.18 Tie’s Extra Questions: Fall 2015### Tie’s Extra Questions: Fall 2015
Let $f ( z ) = \sum _ { { \cal \ L } } ^ { \infty } c _ { n } z ^ { n }$ be analytic and one-to-one in $r _ { 0 } < | z | < R _ { 0 }$ . For $r _ { 0 } < r < R < R _ { 0 }$ , let n=−∞ $D ( r , R )$ be the annulus $r < | z | < R$ . Show that the area of $f ( D ( r , R ) )$ is finite and is given by
$$
S = \pi \sum _ { n = - \infty } ^ { \infty } n | c _ { n } | ^ { 2 } ( R ^ { 2 n } - r ^ { 2 n } ) .
$$
## 20.6 Spring 2015
## 20.6.1 Tie’s Extra Questions: Spring 2015
Let $f _ { n } , f$ be analytic functions on the unit disk D. Show that the following are equivalent.
(i) $f _ { n } ( z )$ converges to f(z) uniformly on compact subsets in D.
(ii) $\int _ { | z | = r } | f _ { n } ( z ) - f ( z ) | | d z |$ converges to 0 if $0 < r < 1$
## 20.6.2 Tie’s Extra Questions: Spring 2015
Let f and g be non-zero analytic functions on a region Ω. Assume $| f ( z ) | = | g ( z ) |$ for all z in Ω. Show that $f ( z ) = e ^ { i \theta } g ( z )$ in Ω for some $0 \leq \theta < 2 \pi$
## 20.6.3 Tie’s Extra Questions: Spring 2015
Suppose f is analytic in an open set containing the unit disc D and $| f ( z ) | = 1$ when $| z | { = } 1$ . Show that either $f ( z ) = e ^ { i \theta }$ for some $\theta \in \mathbb { R }$ or there are finite number of $z _ { k } \in \mathbb { D }$ $k \leq n$ and $\theta \in \mathbb { R }$ such that $f ( z ) = e ^ { i \theta } \prod _ { k = 1 } ^ { n } { \frac { z - z _ { k } } { 1 - { \bar { z } } _ { k } z } }$
Also cf.
Stein et al, 1.4.7, 3.8.17
## 20.6.4 Tie’s Extra Questions: Spring 2015
(1) Let $p ( z )$ be a polynomial, $R > 0$ any positive number, and $m \geq 1$ an integer.
Let $M _ { R } =$ su $\mathfrak { s } \{ | z ^ { m } p ( z ) - 1 | : | z | = R \}$ . Show that $M _ { R } > 1$ .
(2) Let $m \geq 1$ be an integer and $K = \{ z \in \mathbb { C } : r \leq | z | \leq R \}$ where $r < R$ . Show (i) using (1) as well as, (ii) without using (1) that there exists a positive number $\varepsilon _ { 0 } > 0$ such that for each polynomial $p ( z )$ ,
$$
\operatorname* { s u p } \{ | p ( z ) - z ^ { - m } | : z \in K \} \geq \varepsilon _ { 0 } .
$$
## 20.6.5 Tie’s Extra Questions: Spring 2015
Let $f ( z ) = { \frac { 1 } { z } } + { \frac { 1 } { z ^ { 2 } - 1 } }$ . Find all the Laurent series of f and describe the largest annuli in which these series are valid.
## 20.6.6 Tie’s Extra Questions: Spring 2015
Suppose f is entire and there exist A, $R > 0$ and natural number N such that $| f ( z ) | \leq A | z | ^ { N }$ for $| z | \geq R$ . Show that (i) f is a polynomial and (ii) the degree of f is at most N .
## 20.6.7 Tie’s Extra Questions: Spring 2015
Suppose f is entire and there exist A, R > 0 and natural number N such that $| f ( z ) | \geq A | z | ^ { N }$ for $| z | \geq R$ . Show that (i) f is a polynomial and (ii) the degree of f is at least N .
## 20.6.8 Tie’s Extra Questions: Spring 2015
(1) Explicitly write down an example of a non-zero analytic function in $| z | < 1$ which has infinitely zeros in $| z | < 1$
(2) Why does not the phenomenon in (1) contradict the uniqueness theorem?
## 20.6.9 Tie’s Extra Questions: Spring 2015
(1) Assume u is harmonic on open set O and $z _ { n }$ is a sequence in O such that $u ( z _ { n } ) = 0$ and lim $z _ { n } \in O$ . Prove or disprove that u is identically zero.
What if O is a region?
(2) Assume u is harmonic on open set O and $u ( z ) = 0$ on a disc in O. Prove or disprove that u is identically zero.
What if O is a region?
(3) Formulate and prove a Schwarz reflection principle for harmonic functions
cf.
Theorem 5.6 on p.60 of Stein et al.
Hint: Verify the mean value property for your new function obtained by Schwarz reflection principle.
## 20.6.10 Tie’s Extra Questions: Spring 2015
Let f be holomorphic in a neighborhood of $D _ { r } ( z _ { 0 } )$ . Show that for any $s < r$ , there exists a constant $c > 0$ such that
$$
\| f \| _ { ( \infty , s ) } \leq c \| f \| _ { ( 1 , r ) } ,
$$
where $| f | | _ { ( \infty , s ) } = \operatorname* { s u p } _ { z \in D _ { s } ( z _ { 0 } ) } | f ( z ) |$ and $| | f | | _ { ( 1 , r ) } = \int _ { D _ { r } ( z _ { 0 } ) } | f ( z ) | d x d y .$
Note: Exercise 3.8.20 on p.107 in Stein et al is a straightforward consequence of this stronger result using the integral form of the Cauchy-Schwarz inequality in real analysis.
## 20.6.11 Tie’s Extra Questions: Spring 2015
(1) Let f be analytic in $\Omega : 0 < | z - a | < r$ except at a sequence of poles $a _ { n } \in \Omega$ with lim $a _ { n } = a$ n→∞ Show that for any $w \in \mathbb { C } .$ , there exists a sequence $z _ { n } \in \Omega$ such that $\operatorname* { l i m } _ { n \to \infty } f ( z _ { n } ) = { \overset { . . . } { w } }$
(2) Explain the similarity and difference between the above assertion and the Weierstrass-Casorati theorem.
## 20.6.12 Tie’s Extra Questions: Spring 2015
Compute the following integrals.
$$
\begin{array} { l } { i \displaystyle \int _ { 0 } ^ { \infty } \displaystyle \frac { 1 } { ( 1 + x ^ { n } ) ^ { 2 } } d x , n \geq 1 \ \mathrm { ( i i ) } \ \int _ { 0 } ^ { \infty } \displaystyle \frac { \cos x } { ( x ^ { 2 } + a ^ { 2 } ) ^ { 2 } } d x , a \in \mathbb { R } \ \mathrm { ( i i i ) } \ \int _ { 0 } ^ { \pi } \displaystyle \frac { 1 } { a + \sin \theta } d \theta , a > 1 } \\ { i v \displaystyle \int _ { 0 } ^ { \frac { \pi } { 2 } } \displaystyle \frac { d \theta } { a + \sin ^ { 2 } \theta } , a > 0 . \ \mathrm { ( v ) } \displaystyle \int _ { | z | = 2 } \displaystyle \frac { 1 } { ( z ^ { 5 } - 1 ) ( z - 3 ) } d z \ \mathrm { ( v ) } \displaystyle \int _ { - \infty } ^ { \infty } \displaystyle \frac { \sin \pi a } { \cosh \pi x + \cos \pi a } e ^ { - i x \xi } d x , \ 0 < a < 1 , } \\ { \xi \in \mathbb { R } \ \mathrm { ( v i ) } \displaystyle \int _ { | z | = 1 } \cot ^ { 2 } z d z . } \end{array}
$$
## 20.6.13 Tie’s Extra Questions: Spring 2015
Compute the following integrals.
$$
\begin{array} { l } { { \displaystyle { \dot { \ i } } \int _ { 0 } ^ { \infty } { \frac { \sin x } { x } } d x \ { \mathrm { ( i i ) } } \int _ { 0 } ^ { \infty } { \bigl ( } { \frac { \sin x } { x } } { \bigr ) ^ { 2 } } d x \ { \mathrm { ( i i i ) } } \int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { ( 1 + x ) ^ { 2 } } } d x , 0 < a < 2 } \\ { { \dot { \iota } } } \\ { { \dot { \iota } } \int _ { 0 } ^ { \infty } { \frac { \cos a x - \cos b x } { x ^ { 2 } } } d x , a , b > 0 \ { \mathrm { ( i i ) } } \int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { 1 + x ^ { n } } } d x , 0 < a < n } \\ { { \dot { \iota } } { \dot { \iota } } i \int _ { 0 } ^ { \infty } { \frac { \log x } { 1 + x ^ { n } } } d x , n \geq 2 \ { \mathrm { ( i v ) } } \int _ { 0 } ^ { \infty } { \frac { \log x } { ( 1 + x ^ { 2 } ) ^ { 2 } } } d x \ { \mathrm { ( v ) } } \int _ { 0 } ^ { \pi } \log | 1 - a \sin \theta | d \theta , a \in \mathbb { C } } \end{array} }
$$
## 20.6.14 Tie’s Extra Questions: Spring 2015
Let $0 < r < 1$ . Show that polynomials $P _ { n } ( z ) = 1 + 2 z + 3 z ^ { 2 } + \cdot \cdot \cdot + n z ^ { n - 1 }$ have no zeros in $| z | < r$ for all sufficiently large $n \mathrm { { ^ { \circ } s } }$
## 20.6.15 Tie’s Extra Questions: Spring 2015
Let $f$ be an analytic function on a region Ω. Show that $f$ is a constant if there is a simple closed curve $\gamma$ in Ω such that its image $f ( \gamma )$ is contained in the real axis.
## 20.6.16 Tie’s Extra Questions: Spring 2015
(1) Show that $\frac { \pi ^ { 2 } } { \sin ^ { 2 } \pi z }$ and $g ( z ) = \sum _ { n = - \infty } ^ { \infty } \frac { 1 } { ( z - n ) ^ { 2 } }$ have the same principal part at each integer point.
(2) Show that $h ( z ) = \frac { \pi ^ { 2 } } { \sin ^ { 2 } \pi z } - g ( z )$ is bounded on C and conclude that ${ \frac { \pi ^ { 2 } } { \sin ^ { 2 } \pi z } } = \sum _ { n = - \infty } ^ { \infty } { \frac { 1 } { ( z - n ) ^ { 2 } } }$
## 20.6.17 Tie’s Extra Questions: Spring 2015
Let $f ( z )$ be an analytic function on $\mathbb { C } \backslash \{ z _ { 0 } \}$ , where $z _ { \mathrm { 0 } }$ is a fixed point.
Assume that $f ( z )$ is bijective from $\mathbb { C } \backslash \{ z _ { 0 } \}$ onto its image, and that f (z) is bounded outside $D _ { r } ( z _ { 0 } )$ , where r is some fixed positive number.
Show that there exist $a , b , c , d \in \mathbb { C }$ with ad $- b c \neq 0 , c \neq 0$ such that $f ( z ) = { \frac { a z + b } { c z + d } } .$
## 20.6.18 Tie’s Extra Questions: Spring 2015
Assume $f ( z )$ is analytic in $\mathbb { D } : | z | < 1$ and $f ( 0 ) = 0$ and is not a rotation $( \mathrm { i . e . } ~ f ( z ) \neq e ^ { i \theta } z )$ Show that $\sum _ { n = 1 } ^ { \infty } f ^ { n } ( z )$ converges uniformly to an analytic function on compact subsets of D, where $f ^ { n + 1 } ( z ) = f ( f ^ { n } ( z ) )$
## 20.6.19 Tie’s Extra Questions: Spring 2015
Let f be a non-constant analytic function on D with $f ( \mathbb { D } ) \subseteq \mathbb { D }$ . Use $\psi _ { a } ( f ( z ) )$ (where $a = f ( 0 )$ $\psi _ { a } ( z ) = \frac { a - z } { 1 - \bar { a } z } )$ to prove that $\frac { | f ( 0 ) | - | z | } { 1 + | f ( 0 ) | | z | } \leq | f ( z ) | \leq \frac { | f ( 0 ) | + | z | } { 1 - | f ( 0 ) | | z | } ,$
## 20.6.20 Tie’s Extra Questions: Spring 2015
Find a conformal map
1. from $\{ z : | z - 1 / 2 | > 1 / 2 , { \mathrm { R e } } ( z ) > 0 \}$ to H
2. from $\{ z : | z - 1 / 2 | > 1 / 2 , | z | < 1 \}$ to D
3. from the intersection of the disk $| z + i | < { \sqrt { 2 } }$ with H to D.
4. from $\mathbb { D } \backslash [ a , 1 )$ to ${ \mathbb D } \backslash [ 0 , 1 ) ( 0 < a < 1 )$
Short solution possible using Blaschke factor.
5. from $\{ z : | z | < 1 , \mathrm { R e } ( z ) > 0 \} \backslash ( 0 , 1 / 2 ] \mathrm { ~ t o ~ } \mathbb { H }$
## 20.6.21 Tie’s Extra Questions: Spring 2015
Let C and $C ^ { \prime }$ be two circles and let $z _ { 1 } \in C , z _ { 2 } \notin C , z _ { 1 } ^ { \prime } \in C ^ { \prime } , z _ { 2 } ^ { \prime } \notin C ^ { \prime }$ . Show that there is a unique fractional linear transformation $f$ with $f ( C ) = C ^ { \prime }$ and $f ( z _ { 1 } ) = z _ { 1 } ^ { \prime } , f ( z _ { 2 } ) = z _ { 2 } ^ { \prime }$
## 20.6.22 Tie’s Extra Questions: Spring 2015
Assume $f _ { n } \in H ( \Omega )$ is a sequence of holomorphic functions on the region Ω that are uniformly bounded on compact subsets and $f \in H ( \Omega )$ is such that the set $\{ z \in \Omega : \operatorname* { l i m } _ { n \infty } f _ { n } ( z ) = f ( z ) \}$ has a limit point in Ω. Show that $f _ { n }$ converges to f uniformly on compact subsets of Ω.
## 20.6.23 Tie’s Extra Questions: Spring 2015
Let $\psi _ { \alpha } ( z ) = { \frac { \alpha - z } { 1 - \bar { \alpha } z } }$ with $| \alpha | < 1$ and ${ \mathbb D } = \{ z : ~ | z | < 1 \}$ . Prove that
$$
\bullet \ \frac { 1 } { \pi } \iint _ { \mathbb { D } } | \psi _ { \alpha } ^ { \prime } | ^ { 2 } d x d y = 1 .
$$
$$
\bullet \ : \frac { 1 } { \pi } \iint _ { \mathbb { D } } | \psi _ { \alpha } ^ { \prime } | d x d y = \frac { 1 - | \alpha | ^ { 2 } } { | \alpha | ^ { 2 } } \log \frac { 1 } { 1 - | \alpha | ^ { 2 } } .
$$
## 20.6.24 Tie’s Extra Questions: Spring 2015
Prove that $f ( z ) = - \frac { 1 } { 2 } \left( z + \frac { 1 } { z } \right)$ is a conformal map from half disc $\left\{ z = x + i y : ~ | z | < 1 , ~ y > 0 \right\}$ to upper half plane $\mathbb { H } = \left\{ \dot { z } = x + i y : ~ y > 0 \right\}$
## 20.6.25 Tie’s Extra Questions: Spring 2015
Let Ω be a simply connected open set and let γ be a simple closed contour in Ω and enclosing a bounded region U anticlockwise.
Let $f : \ \Omega \to \mathbb { C }$ be a holomorphic function and $| f ( z ) | \leq M$ for all $z \in \gamma$ . Prove that $| f ( z ) | \leq M$ for all $z \in U$
## 20.6.26 Tie’s Extra Questions: Spring 2015
Compute the following integrals.
(i) $\int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { 1 + x ^ { n } } } d x , 0 < a < n { \mathrm { ~ ( i i ) } } \int _ { 0 } ^ { \infty } { \frac { \log x } { ( 1 + x ^ { 2 } ) ^ { 2 } } } d x$
## 20.6.27 Tie’s Extra Questions: Spring 2015
Let $0 < r < 1$ . Show that polynomials $P _ { n } ( z ) = 1 + 2 z + 3 z ^ { 2 } + \cdot \cdot \cdot + n z ^ { n - 1 }$ have no zeros in $| z | < r$ for all sufficiently large $n \mathrm { { ^ { \circ } s } }$
## 20.6.28 Tie’s Extra Questions: Spring 2015
Let f be holomorphic in a neighborhood of $D _ { r } ( z _ { 0 } )$ . Show that for any $s < r$ , there exists a constant $c > 0$ such that
$$
\| f \| _ { ( \infty , s ) } \leq c \| f \| _ { ( 1 , r ) } ,
$$
where $\| f \| _ { ( \infty , s ) } = \operatorname* { s u p } _ { z \in D _ { s } ( z _ { 0 } ) } | f ( z ) |$ and $\| f \| _ { ( 1 , r ) } = \int _ { D _ { r } ( z _ { 0 } ) } | f ( z ) | d x d y .$
## 20.6.29 Tie’s Extra Questions: Spring 2015
Let $\psi _ { \alpha } ( z ) = { \frac { \alpha - z } { 1 - \bar { \alpha } z } }$ with $| \alpha | < 1$ and ${ \mathbb D } = \{ z : ~ | z | < 1 \}$ . Prove that
$$
\bullet \ \frac { 1 } { \pi } \iint _ { \mathbb { D } } | \psi _ { \alpha } ^ { \prime } | ^ { 2 } d x d y = 1 .
$$
$$
\bullet \ : \frac { 1 } { \pi } \iint _ { \mathbb { D } } | \psi _ { \alpha } ^ { \prime } | d x d y = \frac { 1 - | \alpha | ^ { 2 } } { | \alpha | ^ { 2 } } \log \frac { 1 } { 1 - | \alpha | ^ { 2 } } .
$$
Prove that $f ( z ) = - \frac { 1 } { 2 } \left( z + \frac { 1 } { z } \right)$ is a conformal map from half disc $\left\{ z = x + i y : ~ | z | < 1 , ~ y > 0 \right\}$ to upper half plane H $= \{ \dot { z } = x + i y : ~ y > 0 \}$
## 20.6.30 Tie’s Extra Questions: Spring 2015
Let Ω be a simply connected open set and let γ be a simple closed contour in Ω and enclosing a bounded region U anticlockwise.
Let $f : \ \Omega \to \mathbb { C }$ be a holomorphic function and $| f ( z ) | \leq M$ for all $z \in \gamma$ . Prove that $| f ( z ) | \leq M$ for all $z \in U$
## 20.6.31 Tie’s Extra Questions: Spring 2015
Compute the following integrals.
(i) $\int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { 1 + x ^ { n } } } d x , 0 < a < n { \mathrm { ~ ( i i ) } } \int _ { 0 } ^ { \infty } { \frac { \log x } { ( 1 + x ^ { 2 } ) ^ { 2 } } } d x$
## 20.6.32 Tie’s Extra Questions: Spring 2015
Let $0 < r < 1$ . Show that polynomials $P _ { n } ( z ) = 1 + 2 z + 3 z ^ { 2 } + \cdot \cdot \cdot + n z ^ { n - 1 }$ have no zeros in $| z | < r$ for all sufficiently large $n \mathrm { { ^ { \circ } s } }$
## 20.6.33 Tie’s Extra Questions: Spring 2015
Let f be holomorphic in a neighborhood of $D _ { r } ( z _ { 0 } )$ . Show that for any $s < r$ , there exists a constant $c > 0$ such that
$$
\| f \| _ { ( \infty , s ) } \leq c \| f \| _ { ( 1 , r ) } ,
$$
where $\| f \| _ { ( \infty , s ) } = \operatorname* { s u p } _ { z \in D _ { s } ( z _ { 0 } ) } | f ( z ) |$ and $\| f \| _ { ( 1 , r ) } = \int _ { D _ { r } ( z _ { 0 } ) } | f ( z ) | d x d y .$
## 20.7 Fall 2016
## 20.7.1 Tie’s Extra Questions: Fall 2016
Let $u ( x , y )$ be harmonic and have continuous partial derivatives of order three in an open disc of radius $R > 0$
(a) Let two points $( a , b ) , ( x , y )$ in this disk be given.
Show that the following integral is independent of the path in this disk joining these points:
$$
v ( x , y ) = \int _ { a , b } ^ { x , y } ( - \frac { \partial u } { \partial y } d x + \frac { \partial u } { \partial x } d y ) .
$$
(b) {=tex} \hfill
(i) Prove that $u ( x , y ) + i v ( x , y )$ is an analytic function in this disc.
(ii) Prove that $v ( x , y )$ is harmonic in this disc.
## 20.7.2 Tie’s Extra Questions: Fall 2016
(a) $f ( z ) = u ( x , y ) + i v ( x , y )$ be analytic in a domain $D \subset \mathbb { C }$ . Let $z _ { 0 } = ( x _ { 0 } , y _ { 0 } )$ be a point in D which is in the intersection of the curves $u ( x , y ) = c _ { 1 }$ and $v ( x , y ) = c _ { 2 }$ , where $c _ { 1 }$ and $c _ { 2 }$ are constants.
Suppose that $f ^ { \prime } ( z _ { 0 } ) \neq 0$ . Prove that the lines tangent to these curves at z0 are perpendicular.
(b) Let $f ( z ) = z ^ { 2 }$ be defined in $\mathbb { C } .$
(c) Describe the level curves of $\operatorname { R e } ( f )$ and of Im(f ).
(ii) What are the angles of intersections between the level curves $\operatorname { R e } ( f ) = 0$ and $\operatorname { I m } ( f ) ?$ Is your answer in agreement with part a) of this question?
## 20.7.3 Tie’s Extra Questions: Fall 2016
(a) $f : D \to \mathbb { C }$ be a continuous function, where $D \subset \mathbb { C }$ is a domain.Let $\alpha : [ a , b ] \to D$ be a smooth curve.
Give a precise definition of the complex line integral
$$
\int _ { \alpha } f .
$$
(b) Assume that there exists a constant M such that $| f ( \tau ) | \le M$ for all $\tau \in$ Image(α). Prove that
$$
\big | \int _ { \alpha } f \big | \leq M \times \mathrm { l e n g t h } ( \alpha ) .
$$
(c) Let $C _ { R }$ be the circle $| z | = R ,$ described in the counterclockwise direction, where $R > 1$ Provide an upper bound for $| \int _ { C _ { R } } \frac { \log { ( z ) } } { z ^ { 2 } } |$ , which depends only on R and other constants.
## 20.7.4 Tie’s Extra Questions: Fall 2016
(a) Let Let $f : \mathbb { C } \to \mathbb { C }$ be an entire function.
Assume the existence of a non-negative integer $m _ { : }$ and of positive constants L and R, such that for all z with $| z | > R$ the inequality
$$
| f ( z ) | \leq L | z | ^ { m }
$$
holds.
Prove that f is a polynomial of degree $\leq m$
(b) Let $f : \mathbb { C } \to \mathbb { C }$ be an entire function.
Suppose that there exists a real number M such that for all $z \in \mathbb { C }$
$$
\operatorname { R e } ( f ) \leq M .
$$
Prove that f must be a constant.
## 20.7.5 Tie’s Extra Questions: Fall 2016
Prove that all the roots of the complex polynomial
$$
z ^ { 7 } - 5 z ^ { 3 } + 1 2 = 0
$$
lie between the circles $| z | = 1$ and $| z | = 2$
## 20.7.6 Tie’s Extra Questions: Fall 2016
(a) Let F be an analytic function inside and on a simple closed curve C, except for a pole of order $m \geq 1$ at $z = a$ inside C. Prove that
$$
\frac { 1 } { 2 \pi i } \oint _ { C } F ( \tau ) d \tau = \operatorname * { l i m } _ { \tau a } \frac { d ^ { m - 1 } } { d \tau ^ { m - 1 } } \big ( ( \tau - a ) ^ { m } F ( \tau ) \big ) \big ) .
$$
(b) Evaluate
$$
\oint _ { C } { \frac { e ^ { \tau } } { ( \tau ^ { 2 } + \pi ^ { 2 } ) ^ { 2 } } } d \tau
$$
where C is the circle $| z | = 4$
## 20.7.7 Tie’s Extra Questions: Fall 2016
Find the conformal map that takes the upper half-plane comformally onto the half-strip $\{ w =$ $x + i y : \ - \pi / 2 < x < \pi / 2 \ y > 0 \}$
## 20.7.8 Tie’s Extra Questions: Fall 2016
Compute the integral $\int _ { - \infty } ^ { \infty } { \frac { e ^ { - 2 \pi i x \xi } } { \cosh \pi x } }$ dx where cosh $z = { \frac { e ^ { z } + e ^ { - z } } { 2 } }$
## 21 Fall 2021
## 21.1 Fall 2021 #1
Let $f ( z )$ be an analytic function on $| z | < 1$ . Prove that $f ( z )$ is necessarily a constant if $f ( \bar { z } )$ is also analytic.
## 21.2 Fall 2021 #2
Let $\gamma ( t )$ be a piecewise smooth curve in $\mathbb { C } , t \in [ 0 , 1 ]$ . Let $F ( w )$ be a continuous function on $\gamma .$ Show that f(z) defined by
$$
f ( z ) : = \int _ { \gamma } \frac { F ( w ) } { w - z } d w
$$
is analytic on the complement of the curve $\gamma .$
## 21.3 Fall 2021 #3
Suppose $n \geq 2$ . Use a wedge of angle $\frac { 2 \pi } { n }$ to evaluate the integral
$$
I = \int _ { 0 } ^ { \infty } { \frac { 1 } { 1 + x ^ { n } } } d x
$$
## 21.4 Fall 2021 #4
Prove that the sequence $\left( 1 + { \frac { z } { n } } \right) ^ { n }$ converges uniformly to $e ^ { z }$ on compact subsets of $\mathbb { C } .$
Hint: $e ^ { \pi \log w _ { n } } = w _ { n } ^ { n }$ and $e ^ { x }$ is uniform continuous on compact subsets $o f \mathbb { C }$
## 21.5 Fall 2021 #5
Assume f is an entire function such that $| f ( z ) | = 1 \ \mathrm { o n } \ | z | = 1$ . Prove that $f ( z ) = e ^ { i \theta } z ^ { n }$ , where $\theta$ is a real number and n a non-negative integer.
Suggestion: First use the maximum and minimum modulus theorem to show
$$
f ( z ) = e ^ { i \theta } \prod _ { k = 1 } ^ { n } { \frac { z - z _ { k } } { 1 - { \overline { { z _ { k } } } } z } }
$$
if f has zeros.
## 21.6 Fall 2021 #6
Show that if $f : D ( 0 , R ) \to \mathbb { C }$ is holomorphic, with $| f ( z ) | \leq M$ for some $M > 0$ , then
$$
\left| \frac { f ( z ) - f ( 0 ) } { M ^ { 2 } - \overline { { f ( 0 ) } } f ( z ) } \right| \le \frac { | z | } { M R } .
$$
## 21.7 Fall 2021 #7
Find a conformal map from the intersection of $| z - 1 | < 2$ and $| z + 1 | < 2$ to the upper half plane.