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# Complex Analysis Qual Questions
$$
\def\RR{{\mathbb R}}
\def\CC{{\mathbb C}}
\def\DD{{\mathbb D}}
\newcommand{\theset}[1]{\left\{ #1 \right\}}
\newcommand{\abs}[1]{\left| {#1} \right|}
\def\dx{{\, dx}}
\def\dz{{\, dz}}
$$
Links to exams: <https://www.math.uga.edu/complex-analysis-exams>
**Progress**
| Exam | Typeset | Imported to MakeMeAQual | Solutions |
| ----------------------- | --------- | ------------------------- | ----------- |
| Edward Azoff's Collection | ✓ | | |
| Jingzhi Tie's Collection | ✓ | ✓ | |
| Spring 2021 | ✓ | | |
| Fall 2020 | | | |
| Spring 2020 | | | |
| Fall 2019 | | | |
| Spring 2019 | | | |
| Fall 2018 | | | |
| Spring 2018 | | | |
| Fall 2017 | | | |
| Spring 2017 | | | |
| Fall 2016 | | | |
| Spring 2016 | | | |
| Fall 2015 | | | |
| Spring 2015 | | | |
| Fall 2014 | | | |
| Spring 2014 | | | |
# Spring 2021
## Question 1 (Spring 2021)
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$yx+b$
$$f(x) = 2x+3$$
- a
- b
- c
> Topics: complex arithmetic, geometry of $\CC$.
Let $z_1, z_2$ be two complex numbers.
a.
Show that
$$
| z_1 -\bar{z_1}z_2|^2 - |z_1 - z_2|^2 = (1 - |z_1|^2)(1 - |z_2|)
$$
b.
Show that if $|z_1| < 1$ and $|z_2| < 1$, then
$$
\abs{
z_1 -z_2 \over 1 - \overline z_1 z_2
}
< 1
$$
c.
Assume that $z_1 \neq z_2$.
Show that
$$
\abs{
z_1 -z_2 \over 1 - \overline z_1 z_2
} < 1
\iff
|z_1| = 1 \text{ or } |z_2| = 1
$$
## Question 2 (Spring 2021)
> Topics: integrals, residues.
Let $\xi\in \RR$ and $\cosh(x) := {1\over 2}(e^x + e^{-x})$.
Evaluate the integral
$$
\int_{-\infty}^\infty {e ^{i\xi x} \over \cosh(x) } \dx
$$
*Hint: use an appropriate rectangular contour where one side is $[-R, R]$.*
## Question 3 (Spring 2021)
> Topics: Liouville? Power series?
Suppose $f$ is entire and there exist $A, R> 0$ and $N\in \Bbb{N}$ such that
$$
|f(z)| \geq A |z|^N \quad\text{for } |z| \geq R.
$$
Show that
a.
$f$ is a polynomial, and
b.
The degree of $f$ is at least $N$.
## Question 4 (Spring 2021)
> Topics: power series?
Let $f(z) = u+iv$ be an entire function such that $u(x, y) = \mathrm{Re}(f(x+iy))$ is a polynomial in $x$ and $y$. Show that $f(z)$ is a polynomial in $z$.
## Question 5 (Spring 2021)
> Topics: linear fractional transformations.
Let $f: \Bbb{D} \to \Bbb{D}$ be a holomorphic map of the open unit disk into itself. Show that for any two points $z, w\in \Bbb{D}$,
$$
\abs{
f(w) - f(z) \over 1 - \overline{f(w)} f(z)
}
\leq
\abs{
w - z \over 1 - \overline w z
},
$$
and the inequality is strict for $z\neq w$ unless $f$ is a linear fractional transformation.
## Question 6 (Spring 2021)
> Topics: integrals, convergence.
Supose $\theset{f_n}_{n=1}^\infty$ is a sequence of holomorphic functions on the disc $\DD$ and $f$ is a holomorphic function on $\DD$. Show that the following are equivalent:
a.
$\theset{f_n}_{n=1}^\infty$ converges to $f$ uniformly on compact subsets of $\DD$.
b.
If $0<r<1$,
$$
\int_{\abs z = r} \abs{ f_n(z) - f(z) } \abs{ \dz } \quad\overset{n\to\infty}\longrightarrow \quad
0
$$
## Question 7 (Spring 2021)
> Topics: conformal maps
Let $R$ be the intersection of the right half plane and the outside of the circle $\abs{z - {1\over 2}} = {1\over 2}$ with the line segment $L = [1, 2]$ removed, i.e.
$$
R := \theset{
z : \quad
\mathrm{Re}(z) > 0, \,
\abs{z - {1\over 2}} > {1\over 2}
}
\setminus
\theset{
z = x+iy:\quad
1\leq x\leq 2, \,
y=0
}
$$
Find a conformal map from $R$ to the upper half plane.