\documentclass[12pt]{amsart}
\title{Problems in Complex Analysis}
\author{Sachin Gautam}

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\begin{document}
\maketitle

\prob{1} Let $U\subset\C$ be an open set. Prove that $U$ is connected
if, and only if $U$ is path--connected. In this case, show that
any two points of $U$ can be joined by a {\em zig--zag} path, i.e.,
a path consisting only of horizontal and vertical line segments.\\

\prob{2} Let $n\in\Z_{\geq 2}$ and $z_1,\ldots,z_n,w_1,\ldots,w_n\in\C$.
Prove the following identity:
\[
\left| \sum_{k=1}^n z_k\ol{w_k}\right|^2 = 
\left(\sum_{k=1}^n |z_k|^2\right)\left(\sum_{k=1}^n |w_k|^2\right)
- \sum_{1\leq k<\ell\leq n} |z_kw_\ell-z_\ell w_k|^2\ .
\]
{\em This identity is due to Lagrange. It immediately implies
the Cauchy--Schwarz inequality.}\\

\prob{3} Let $z_1,z_2,z_3\in S^1 = \{z:|z|=1\}$ be such that
$z_1+z_2+z_3=0$. Show that these three points form vertices of an equilateral
triangle.\\

\prob{4} Prove the {\em Ptolemy relation}: if $A,B,C,D$ are four
distinct points on a circle, then
\[
|AC| |BD| = |AB| |CD| + |AD| |BC|\ ,
\]
where $|PQ|$ denotes the length of the line segment joining $P$ and $Q$.\\

\prob{5} Assume $z_1,z_2\in\C$ are two distinct points.
Sketch the following curves as $\lambda$ varies over
$\R_{>0}$ (circles of Apollonius)
\[
\left\{ z\in\C : |z-z_1| = \lambda|z-z_2| \right\}.
\]

\prob{6} Sketch the following curves for $\lambda\in\R_{>0}$
(Bernoulli's lemniscate is when $\lambda=1$): $|z^2-1|=\lambda$.\\

\prob{7} Let $g\in C^2(\Omega;\R)$, where $\Omega\subset\C$ is an
open and connected set. Show that the Laplace equation for $g$, written
in polar coordinates, takes the following form:
\[
r^2\frac{\partial^2g}{\partial r^2} + r \frac{\partial g}{\partial r}
+ \frac{\partial^2g}{\partial \theta^2} = 0\ .
\]
Use this to prove that if $g$ is harmonic, and independent of $\theta$,
then $g = A\ln(r)+B$, for some constants $A,B\in\R$.\\

\prob{8} Let $n\in\Z_{\geq 2}$. Describe the level curves
$\Re(z^n) = 0$ and $\Im(z^n)=0$.\\

\prob{9} Show that $\ds u(x,y) = \frac{x}{x^2+y^2}$ is harmonic.
Compute its harmonic conjugate (the domain is $\nC = \C\setminus\{0\}$).\\

\prob{10} Let $a\in [0,1)$ and $r\in [0,1-a)$. Let $C(a;r)$ be the
circle centered at $a$ of radius $r$, and $S^1 = C(0;1)$ (note:
$C(a;r)$ lies inside $S^1$). Let $A_1=e^{\iota\psi_1}$
and $A_2=e^{\iota\psi_2}$ be two points on $S^1$ such that
the line segment $A_1A_2$ is tangent to $C(a;r)$. What is the
relation between $\psi_1$ and $\psi_2$?
{\em This calculation is due to Jacobi, who used it to prove
Poncelet's porism for circles.}\\

\prob{11} Let $\Omega\subset\C$ be an open, connected and simply--connected
set, $u:\Omega\to\R$ a harmonic function (i.e., $u\in C^2(\Omega,\R)$,
and $u_{xx}+u_{yy}=0$). Show that there exists $v:\Omega\to\R$ such
that $f=u+\iota v$ is holomorphic. Hence, $u$ is real--analytic.
This is an instance of ``regularity theorems" in mathematics.\\

\prob{12} Compute the following integrals.
\begin{enumerate}
\item $\ds \int_{C} \frac{ze^z}{(z-a)^{n+1}}\, dz$. Here
$C=C(a,r)$ is the counterclockwise circle around $a\in\C$ of
radius $r\in\R_{>0}$, and $n\in\Z_{\geq 0}$.

\item $\ds \int_{C} \frac{\sin(z)}{z(1-z)^3}\, dz$. Here
$C=C(0,1/2)$ is the circle of radius $1/2$ around $0$.\\
\end{enumerate}

\prob{13} Let $p(z)=z^n+\sum_{j=0}^{n-1} a_j z^{n-j} \in \C[z]$
be a polynomial of degree $n\geq 2$. Let $z_1,\ldots,z_n\in\C$
be its roots (not necessarily distinct, listed according to their
multiplicity). Let $\zeta\in\C$ be a root of $p'(z)$ such that
$\zeta\not\in\{z_1,\ldots,z_n\}$. Show that
\[
\left(\sum_{j=1}^n \frac{1}{|\zeta-z_j|^2}\right) \zeta = 
\sum_{j=1}^n \frac{z_j}{|\zeta-z_j|^2}\ .
\]
This result is called Gauss--Lucas Theorem. It shows that
$\zeta$ can be written as a linear combination of $z_j$
such that the coefficients are positive real and add up to
$1$.\\

\prob{14} Let $f:\C\to\C$ be a holomorphic function such that
$f(0)=0$ and $\lim_{z\to\infty} \Re(f(z)) = 0$. Show that
$f(z)=0$ for every $z\in\C$. (Hint: show that $e^f$ is
bounded, and use Liouville's theorem).\\

\prob{15} {\em Dirichlet's test.} Let $\{a_n\}_{n=1}^{\infty}$
and $\{b_n\}_{n=1}^{\infty}$ be two sequences of complex numbers.
Set $S_k = a_1+\cdots+a_k$ ($k\geq 1$). Show that:
\begin{enumerate}
\item For every $n,m\in\Z_{\geq 1}$, $n\geq m$, we have:
\[
\sum_{k=m}^n a_kb_k = \sum_{k=m}^{n-1} S_k(b_k-b_{k-1})
- S_{m-1}b_m + S_nb_m\ .
\]
This formula is called {\em Abel's transformation}.

\item Assume that there exists $M\in\R_{>0}$ such that
$|S_k|\leq M$, for every $k\geq 1$. Further, assume
that $\{b_n\}$ is real, monotonically non--increasing,
and $\lim_{n\to\infty} b_n = 0$. Show that
$\sum_{n=1}^\infty a_nb_n$ is convergent.\\

\end{enumerate}

\prob{16} {\em Weierstrass' M test.} Let $\{a_n(z)\}_{n=1}^{\infty}$
be a sequence of functions defined on a domain $\Omega$.
Assume that, for every compact set $K\subset\Omega$, there
exists $M_K(n)\in\R_{>0}$ such that
\begin{itemize}
\item $|a_n(z)|<M_K(n)$ for every $z\in K$ and $n\geq 1$.
\item $\sum_{n=1}^{\infty} M_k(n)$ is convergent.
\end{itemize}
Then, $\sum_{n=1}^{\infty} a_n(z)$ converges, uniformly and
absolutely, on $\Omega$.\\

\prob{17} Decide whether the following series are convergent or not.
For the second, decide the values of $\theta$ for which it converges.
\[
\text{(a) } \sum_{n=1}^{\infty} e^{\iota n}\,,\qquad
\text{(b) } \sum_{n=1}^{\infty} \frac{e^{\iota n\theta}}{n}\,.
\] 

\prob{18} Let $\sum_{n=0}^{\infty} a_nz^{n+1}$ be a power series with
non--zero radius of convergence. Show that $\ds\sum_{n=0}^{\infty} a_n\frac{t^n}{n!}$
has infinite radius of convergence.\\

\prob{19} {\em Bernoulli numbers.} Define $\{B_n\}_{n=0}^{\infty} \subset \Q$
by:
\[
\frac{z}{e^z-1} = \sum_{n=0}^{\infty} B_n \frac{z^n}{n!}, \text{ for } |z|<2\pi.
\]
\begin{itemize}
\item[(a)] Show that $B_0=1$, $B_1=-\frac{1}{2}$ and $B_{2k+1}=0$, for all $k\geq 1$.
\item[(b)] Prove the following recurrence relation for $\{B_n\}$:
\[
B_{n} = -\frac{1}{n+1} \sum_{\ell=0}^{n-1} \left(\begin{array}{c} n+1\\ \ell
\end{array}\right) B_\ell\, \forall\ n\geq 1.
\]
\item[(c)] Write the Taylor series expansion of $\log(\sin(z)/z)$, near $z=0$,
and indicate its radius of convergence.\\
\end{itemize}

\prob{20} Let $\ds A(z)=\sum_{k=0}^{\infty} a_k z^k$. Prove that $A(z)$ is
the Taylor series expansion, near $0$, of a rational function (defined
at $0$) if, and only if there exist $q_1,\ldots,q_\ell\in\C$
and $N\geq \ell$ such that
\[
a_n = q_1a_{n-1} + q_2a_{n-2} + \cdots + q_\ell a_{n-\ell},\ \text{for every }
n\geq N.
\]

\prob{21} Let $\ds f(z)=\sum_{n=0}^{\infty} a_nz^n$ be the Taylor series
of a holomorphic function, for all $z\in D(0;R)$.
\begin{itemize}
\item[(a)] Show that for every $r<R$, we have:
\[
\frac{1}{2\pi} \int_0^{2\pi} |f(re^{\iota\theta})|^2\, d\theta
=
\sum_{n=0}^{\infty} |a_n|^2 r^{2n}\ .
\]
\item[(b)] For each $r<R$, let $M(r) := \text{Max}\{|f(s)|:|s|=r\}$.
Prove that $\ds |c_n|\leq \frac{M(r)}{r^n}$, for every $n\geq 0$.

\item[(c)] Show that, if there exists $r\in (0,R)$ and $n\in\Z_{\geq 0}$
such that $|c_n|r^n = M(r)$, then $f(z) = c_n z^n$.\\
\end{itemize}

\prob{22} For a holomorphic function $f:\Omega\to\C$ and an isolated
singularity $\alpha$ of $f$ (i.e., $D^{\times}(\alpha,r)\subset\Omega$
for some $r>0$), define:
\[
\Res(f;\alpha) := \frac{1}{2\pi\iota} \int_{C(\alpha,r)} f(z)\, dz\,.
\]

\begin{itemize}
\item[(a)] If $\ds f(z) = \sum_{m=1}^{\infty} d_m (z-\alpha)^{-m}
+ \sum_{n=0}^{\infty} c_n (z-\alpha)^n$ is the Laurent series expansion
of $f$ on the punctured neighbourhood $D^{\times}(\alpha,r)$, then
show that $\ds\Res(f;\alpha) = d_1$.

\item[(b)] Let $\varphi:D(\alpha,r)\to\C$ be another, holomorphic function.
Show that $\Res(\varphi f; \alpha) = \varphi(\alpha)\Res(f;\alpha)$.

\item[(c)] Same $\varphi$ as above, assume that $\alpha$ is a pole of
order $N$, for the function $f$. Show that:
\[
\Res\left( \varphi \frac{f'}{f}; \alpha\right) = N\varphi(\alpha)
\]
\end{itemize}

\prob{23} For each of the following, determine a formal solution
$w(z)$, near $z=0$, with $w(0)=0$ and $w'(0)=1$ initial conditions.
Discuss the radius of convergence of your solution.
\begin{itemize}
\item[(a)] $w'' - z^2 w = 3z^2-z^4$.

\item[(b)] $(1-z^2)w'' - 2zw' + n(n+1)w=0$, here $n\in\Z$.\\
\end{itemize}

\prob{24} Let $X$ be an $n\times n$ matrix over $\C$, and let
$\sigma(X) = \{\lambda_j\}_{j=1}^n\subset\C$ be the eigenvalues of $X$
(repeated according to their multiplicities).

\begin{enumerate}
\item Consider the operator $\ad(X):Y\mapsto XY - YX$ on the vector
space of $n\times n$ matrices. Show that the eigenvalues of
$\ad(X)$ are $\{\lambda_j-\lambda_k:1\leq j,k\leq n\}$.\\

\item Let $f:\Omega\to\C$ be a holomorphic function, where $\Omega\subset\C$
is an open, connected set containing $\sigma(X)$.
Define
\[
f(X) := \frac{1}{2\pi\iota} \int_C (z-X)^{-1} f(z)\, dz\,,
\]
where $C$ is a contour in $\Omega$ containing $\sigma(X)$. Show that
the eigenvalues of $f(X)$ are $\{f(\lambda_j):1\leq j\leq n\}$.\\

\item Let $|\!|\cdot |\!|$ denote a norm on the space of
$n\times n$ matrices. Prove the following (Gelfand's formula)
\[
\lim_{n\to\infty} |\!| X^n |\!|^{\frac{1}{n}} = \text{Max}\{|\lambda_j| : 1\leq j\leq n\}.
\]
\end{enumerate}

\prob{26} Let $f:\R_{>0}\to\C$ be a continuous function
such that (i) $f(t)\to 0$ as $t\to 0^+$, and (ii) there exist
constants $M,C,R\in\R_{>0}$ such that $|f(t)|<Me^{Ct}$ for all $t>R$.
Show that, for $z\in\C$ such that $\Re(z)>0$, we have:
\[
\int_0^\infty f(t)e^{-zt}\, dt = z^{-1} \int_0^\infty
f'(t)e^{-zt}\, dt.
\]

Use this to prove the identity (Euler): $\ds \int_0^\infty
\frac{t^n}{n!}e^{-zt}\, dt = z^{-n-1}$, for all $n\in\Z_{\geq 0}$.\\

\prob{27} Compute the asymptotic expansions of the following
integrals, as $\Re(z)\to\infty$.
\[
\text{(a) } \int_0^\infty \frac{e^{-zt}}{1+t^n}\, dt\ (n\in\Z_{\geq 1}),
\quad \text{ (b) } \int_0^\infty (1+t)^a e^{-zt}\, dz\ (a\in\C).
\]


\prob{28} Let $g:\C\dashrightarrow\C$ be a meromorphic function,
with finitely many poles at $a_1,\ldots,a_n\in\C$. Let $A\in\R$
be such that $\Re(a_j)<A$ for all $1\leq j\leq n$.
Assume that $\ds\lim_{\begin{subarray}{c} z\to\infty \\ \Re(z)\leq A
\end{subarray}} |g(z)| = 0$. Prove that, for every $t\in\R_{>0}$,
we have:
\[
\int_{L_A} g(z)e^{zt}\, dz = 2\pi\iota
\sum_{k=1}^n \Res_{z=a_j}(f(z)e^{zt}).
\]
Here, $L_A = A+\iota\R$ is the infinite vertical line.\\

\prob{29} Let $a\in\R_{>0}$ and $L_a = a+\iota\R$. Show that
for every $n\in\Z_{\geq 0}$ and $t\in\R_{>0}$, we have:
\[
\frac{1}{2\pi\iota} \int_{L_a} e^{zt}z^{-n-1}\, dz = \frac{t^n}{n!}\ .
\]
{\bf (Bonus)}
Generalize this to obtain the inverse to the Laplace transform.\\

\prob{30} Compute the Laplace transform $\ds\int_0^\infty \varphi(t)e^{-zt}\, dt$
for $\varphi:\R_{>0}\to\R_{>0}$ given by
$\varphi(t) = n$ for $t\in [n,n+1)$.\\

\prob{31} Let $\psi(x+\iota y)=u(x,y)+\iota v(x,y)$ be a holomorphic function on
an open, connected set $\Omega\subset\C$. Let $z_0=x_0+\iota y_0\in\Omega$
be a critical point of $\psi$ (i.e., $\psi'(z_0)=0$)
and let $n\in\Z_{\geq 2}$
be such
that $0=\psi'(z_0)=\ldots=\psi^{(n-1)}(z_0)$ and $\psi^{(n)}(z_0)=ae^{\iota\alpha}
\neq 0$. 

\begin{enumerate}
\item Write $z=x+\iota y = z_0 + \rho e^{\iota\theta}$, $u=u(x,y)$,
$v=v(x,y)$, $u_0 = u(x_0,y_0)$ etc. Show that
\[
u^2+v^2 = u_0^2+v_0^2+\frac{2a\rho^n}{n!} \left(
u_0\cos(n\theta+\alpha)+v_0\sin(n\theta+\alpha)
\right) + O(\rho^{n+1}),\ \ \text{ as } \rho\to 0^+\ .
\]

\item Prove that, through $z_0$, the directions of steepest ascent/descent
are given as follows.\\

\begin{center}
\begin{tabular}{|r|c|l|}
\hline
Steepest descent & $-\frac{\alpha}{n} + (2p+1)\frac{\pi}{n}$ & $p=0,\ldots,n-1$ \\
Steepest ascent & $-\frac{\alpha}{n} + 2p\frac{\pi}{n}$ & $p=0,\ldots,n-1$\\
\hline
\end{tabular}
\end{center}

\vspace*{0.1in}

\item Prove the maximum modulus principle: if $|\psi(z)|$ takes its
maximum value at $z_0\in\Omega$, then $\psi$ is a constant function.\\

\end{enumerate}

\prob{32} Compute the critical points and directions of steepest
ascent/descent from each of these points, for the following functions:
(a) $\ds \frac{z^2}{2}- \alpha z$, ($\alpha\in\C$), and
(b) $\ds \frac{z^3}{3} - z$.\\

\prob{33} Prove the Riemann-Lebesgue Lemma: if $(a,b)\subset\R$ and
$f\in L^1(a,b)$ (that is, $\ds \int_a^b|f(t)|\, dt < \infty$), then
\[
\lim_{\lambda\to+\infty}\int_a^b f(t)e^{\iota\lambda t}\, dt
=0\,.
\]

\prob{34} Prove the following asymptotic expansion, as $x\to\infty$:
\[
\int_0^1 \ln(t)e^{\iota x t}\, dt \sim
\iota\frac{\ln(x)}{x} - \frac{\iota\gamma+\pi/2}{x} + 
\iota e^{\iota x}\sum_{n=1}^{\infty} (-1)^n \frac{(n-1)!}{x^{n+1}}\ .
\]
Here, $\gamma$ is the Euler--Mascheroni constant.\\

\prob{35} Discuss the $n\to \infty$ behaviour of $\Psi^{(n)}(1)$,
where $\ds \Psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}$ is the
logarithmic derivative of the gamma function.\\

\prob{36} Assume that $\{a_n\}_{n=0}^{\infty}\subset\C$ is a sequence
of complex numbers such that $f(z) = \sum_{n=0}^{\infty} a_n z^n$
is an entire (i.e., radius of convergence is infinity) function.
Note that, by Cauchy's formula:
\[
a_n = \frac{1}{2\pi\iota} \int_C z^{-n-1} f(z)\, dz
= \frac{1}{2\pi\iota} \int_C e^{-\ln(z)(n+1)} f(z)\, dz\ .
\]
Use this observation, together with the method of steepest
descent, to obtain $n\to \infty$ behaviour of $a_n$, in the following
examples.\\

\noindent (a) $f(z)=e^z$, (b) $f(z) = e^{z^k}$, (c) $a_n = \frac{B_n}{n!}$,
where $B_n$ is the $n$-th Bernoulli number (see Problem 19 above).\\

\prob{37} Discuss the change in asymptotic expansions as $\lambda\to\infty$
of the following integral, as $\theta$ varies in $(0,2\pi)$, $\theta\neq \pi$.
\[
I(\lambda,\theta) = \int_0^{e^{\iota\theta}} \frac{e^{\lambda z^2}}{1+z}\, dz\,.
\]

\prob{38} Compute the leading term behaviour of the following
integral, as $\lambda\to\infty$. Here $a\in\R$ is a fixed
constant, and $C$ is the infinite horizontal line $\R + \iota$.
\[
I(\lambda,a) = \int_C \frac{e^{\iota\lambda \left(\frac{z^3}{3} - z\right)}}{z-a}\, dz\,.
\]
\end{document}
