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\title{Analysis problem set -- Integration workshop 2016}
%\author{}
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\begin{document}
\maketitle
%\section{}
%\subsection{}

\newcounter{excount}



\section{Sequences and series}

\begin{enumerate}

\item Let $x_n$ be a bounded sequence of real numbers. 
Let $x=\limsup_{n \ra \infty} x_n$. 
\begin{enumerate}
\item Prove that there is a subsequence $x_{n_k}$ such that 
$\lim_{k \ra \infty} x_{n_k} = x$. 
\item Prove that for every subsequence $x_{m_k}$ of $x_n$ which
converges, $\lim_{k \ra \infty} x_{m_k} \le x$ 
\end{enumerate}


%\item
%Given a set $S$ in a metric space, let $\bar{S}$ denote 
%the set obtained by adding all the limit points of $S$ to S, 
%i.e., $p \in \bar{S}$ if there is a sequence
%$x_n \in S$ such that $x_n \ra p$. 
%Show that $\bar{\bar{S}} = \bar{S}$. 
%\medskip


%(b) Show that the set of points of discontinuity of an arbitrary real
%valued function is always a $F_\sigma$ set and the set of points of continuity is
%always a $G_\delta$ set.
%
%(c) [Optional.. hard..] There is no real valued function that is continuous
%on all the rationals but discontinuous on all the irrationals (Use
%Baire category theorem).
%\medskip

\item
Assume that $a_n > 0$ and $b_n > 0$ for n = 1, 2, . . ., and suppose that 
$\lim_{n \ra \infty} a_n/b_n =1$. Prove that 
$\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty b_n$
converges.

\item Show that the product $\displaystyle{\prod_n (1+a_n)}$ converges (resp. converges absolutely) if and only if the series $\displaystyle{\sum a_n}$ converges (resp. converges absolutely).



\item A non-negative sequence $a_n$ is {\em subadditive} if $a_{n+m} \leq a_n + a_m$ for all $m,n \in \mathbb{Z}^+$. The goal of this problem is to show that, for any subadditive sequence, the limit $\displaystyle{\lim_{n \to \infty} \frac{a_n}{n} }$ exists.

(a) Show that, for any $n \in \mathbb{Z}^+$, there is a constant $C_n$ such that, for all $k \geq n+1$, we have 
$$
\frac{a_k}{k} \leq \frac{a_n}{n} + \frac{C}{k}.
$$

(b) Show that $\displaystyle{ \limsup_{k \to \infty} \frac{a_k}{k} \leq  \liminf_{n \to \infty} \frac{a_n}{n}}$. Why is this enough to prove the claim?
 
 
\item $A_n$ is a sequence of subsets of $\reals$. 

(a) Explain why the following definitions make sense:
$$
\limsup_n A_n = \bigcap_{j= 1}^\infty \bigcup_{n= j}^\infty A_n = \{x | x \in A_n \mbox{ \em infinitely often}\}
$$
and 
$$
\liminf_n A_n = \bigcup_{j= 1}^\infty \bigcap_{n= j}^\infty A_n = \{x | x \in A_n \mbox{ \em eventually}\}
$$

(b) What can you conclude about a sequence of subsets for which $\limsup_n A_n = \liminf_n A_n$. Explain why this is a very restrictive definition of a limit for sets. In your classes you will explore alternative (and less restrictive) notions for limits of sets.

\item
(a) Let $f$ be a continuous function on $[0,1]$. 
Show that the following limits exist and evaluate them:
\beann
\lim_{n \ra \infty} \int_0^1 \, x^n \, f(x) \, dx
\eeann
\beann
\lim_{n \ra \infty} n \int_0^1 \, x^n \, f(x) \, dx
\eeann

(b) Let $g$ be a differentiable function on $[0,1]$ such that $g(1)=0$. 
Show that the following limit exists and evaluate:
\beann
\lim_{n \ra \infty} n^2 \int_0^1 \, x^n \, g(x) \, dx
\eeann
\medskip

\item Consider the alternating series
$$
1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
$$
(a) Show that, as presented the series (i.e the sequence of partial sums taken with the given ordering) converges to $\ln(2)$.\\
(b) Show that, given any pair of real numbers $a < b$, the terms in the series can be rearranged so that the sequence $x_n$ of partial sums (of the rearranged series) satisfies 
$$
\limsup_n x_n \geq b, \qquad \liminf_n x_n \leq a.
$$

\item Let 
$$
f(x) = \begin{cases} a & x = 0 \\ x^{-x} & x > 0 \end{cases}
$$
(a) Show that there is an appropriate choice for $a$ so that $f$ is continuous on $[0,1]$.

(b) For this choice of $a$, show, with justification, that 
$$
\int_0^1 f(x) dx = \sum_{n = 1}^\infty n^{-n}.
$$
Possibly useful integral: If $\beta > 0$ and $n \in \mathbb{N}$, 
$$
\int_0^\infty x^n e^{-\beta x} dx = \frac{n!}{\beta^{(n+1)}} 
$$  
\item
Give a counter example to the following statement: A sequence of
differentiable functions $f_n$ converges uniformly to a 
differentiable function $f$. 
This implies that the sequence of derivatives $f_n^\prime$
converges pointwise to $f^\prime$.
\medskip

\item
$f_n$ is a sequence of uniformly bounded non-negative Riemann integrable
functions. For each $x \in [0,1]$, the real sequence $f_k(x)$ is monotone non-decreasing, i.e 
$n \ge m$ and $x \in [0,1]$ implies that $f_n(x) \ge f_m(x)$, 
and $0 \le f_n(x) \le  K < \infty$ for all $n, x$.

(a) Show that the sequence $f_n(x)$ converges pointwise to a bounded
function $f(x)$.

(b) Show that the sequence of real numbers 
$s_k = \int_0^1 \, f_k(x)\, dx$ also converges to some number $s$.

(c) Is it true that $f$ is Riemann integrable, and
$\int_0^1 f(x) \, dx = s$ ?
Prove or give a counter example.
\medskip


\end{enumerate}


\section{Metric spaces and continuous functions}
\begin{enumerate}

\item \label{prob:boundedmetric} For $x,y \in \reals$ let 
\beann
d(x,y) = \frac{|x-y|  }{ 1 + |x-y|}
\eeann
\begin{enumerate}
\item Show this is a metric.
\item Does this metric give $\reals$ a different topology from the one that
comes from the usual metric on $\reals$? You should prove your answer. 
\end{enumerate}
  
\item {\bf The structure of open sets in $\mathbb{R}$.}

In what follows, we are considering the standard (metric) topology on 
$\mathbb{R}$.

\begin{enumerate}

\item Let $S$ be a nonempty open subset of $\R$. For each $x \in S$, 
let $A_x = \{a \in \R : (a,x] \subseteq S\}$ and 
$B_x = \{b \in \R : [x,b) \subseteq S\}$. Show that, $A_x$ and 
$B_x$ are both non-empty.

\item Where $x \in S$ as above, if $A_x$ is bounded below, let 
$a_x = \inf(A_x)$. Otherwise, let $a_x = - \infty$, and define 
$b_x$ is a corresponding manner. Show that $x \in I_x = (a_x,b_x) \subseteq S$.

\item Show that $S = \cup_x I_x$.

\item Show that the intervals $I_x$ give a partition of $S$, {\em i.e.}, 
for $x, y \in S$, either $I_x = I_y$ or $I_x \cap I_y = \emptyset$.

\item Show that the set of distinct intervals $\{I_x : x \in S\}$ is countable. 

\item Prove that every open set in $\R$ is a countable disjoint union of 
open intervals.

\end{enumerate}

 
\item If $f:(X,\mathcal{T}) \rightarrow (Y,\mathcal{S})$ and
  $g:(Y,\mathcal{S}) \rightarrow (Z,\mathcal{V})$ are continuous, show
  that the composition $g \circ f :(X,\mathcal{T}) \rightarrow
  (Z,\mathcal{V})$ is also continuous.
  
\item A subset $A$ of $X$ is called dense if the closure of $A$ is $X$. 
A topological space $X$ is called {\em separable} 
if there exists a countable dense subset. 

\begin{enumerate}
\item Prove that $\reals^n$ with the usual topology is separable. 



\item Since the rationals are a dense set in $\reals$ does it follow that every open subset of $\reals$ is determined by the rational elements of the set?

\item Show that the collection of all intervals of the form $(r_1,r_2)$ where both $r_1$ and $r_2$ are rational is countable, and further, every open subset of $\reals$ is uniquely determined by the intervals with rational endpoints that are contained in it.

\item Is the collection of all open subsets of $\reals$ countable?

\item %Recall the definition of $l^p(\naturals)$ with $1 \le p \le \infty$.  For what $p$ is this metric space separable? Prove your answer. 
If a metric space is separable, show that there is a countable collection of open balls (i.e sets of the form $B_\epsilon(x) = \{y: d(x,y) < \epsilon\}$) such that every open set can be written as a union of balls from this collection.
  
\end{enumerate}

\item {\bf Accumulation points}  $A \subseteq \R$, and $A'$ denotes the 
set of all the accumulation points of $A$.
\begin{enumerate}
\item If $y \in A'$ and $U \subseteq \R$ is an open set containing $y$, 
show that there are infinitely many distinct points in $A \cap U$.
\item Show that 
$$
A' = \bigcap_{x \in A} \cl(A\backslash\{x\}).
$$
\item Using this, or otherwise, show that $A'$ is a closed set.
\item Show that $\cl(A) = A \cup A'$.
\end{enumerate}

%%\begin{enumerate}
%%\item 
%$(X,T)$ is a topological space and the topology $T$ is Hausdorff. $A \subseteq X$. Let $A'$ denote the set of accumulation points of $A$. Show that 
%$$
%A' =  \bigcap_{x \in A} \overline{A \backslash\{x\}}.
%$$
%\end{enumerate}

%\item {\bf Boundary points}
%%\begin{enumerate}
%\item Problem 3.5.30, Pg. I-150 of Prof. Flaschka's notes.
%\item Show that the above definition of boundary point agrees with the definition in class, {\em viz.}, $x \in X$ is a boundary point of $A$ iff for every neighborhood $U$ of $x$, $U \cap A \neq \emptyset$ and $U \cap A^c \neq \emptyset$.
%\end{enumerate}



\item 
\begin{enumerate}
\item
Show that $\R$ and $(0,1)$ are homeomorphic, i.e., there is  continuous
bijection between them whose inverse is also continuous.

\item
Let $f : (0,1) \rightarrow \R$ be your homeomorphism. Show there 
is a Cauchy sequence $x_n$ in $(0,1)$ such that $f(x_n)$ is not 
Cauchy in $\R$. 
\end{enumerate}

\item In the lectures we stated a proposition that said that if 
$(X,d)$ and $(Y,d^\prime)$ are metric spaces and $f:X \rightarrow Y$,
then the $\epsilon-\delta$ definition of continuity of $f$ and the open set 
defintion are equivalent. Prove this proposition.

\item Let $C((0,1))$ be the set of bounded continuous functions on 
$(0,1)$ with the usual sup norm. So 
\beann
d(f,g) = \sup_{0 < x < 1} |f(x)-g(x)|
\eeann
Let $U=\{f: f(x) > 0 \quad \forall x \in (0,1)\}$, 
and $V=\{f: f(x) \ge 0 \quad \forall x \in (0,1)\}$. 
For each of $U$ and $V$ determine if the set is open or closed or 
neither. You should prove your answer. 

\item
(a) Define a function $f$ on $[0,1]$ by
$$
f(x) = \begin{cases} 0 & x \mbox{ is irrational} \\ 1 & x \mbox { is rational} \end{cases}
$$
Identify the points where $f$ is continuous and the points where $f$ is discontinuous.

(b) Define a function $g$ on $[0,1]$ by
$$
g(x) = \begin{cases} 0 & x \mbox{ is irrational} \\ \frac{1}{q} & x = \frac{p}{q} \mbox{ expressed in its lowest terms} \end{cases}
$$
Identify the points where $f$ is continuous and the points where $f$ is discontinuous.

(c) (Optional.. hard] Is there an example of a function $f : [0; 1] \ra \reals$ that is 
discontinuous at all the irrationals and continuous at all the rationals?

\item A function $f:\reals \to \reals$ is {\em uniformly continuous} if for all $\epsilon > 0$, there is a $\delta > 0$ such that {\em for all } $ x \in \reals$, $|y-x| < \delta \implies |f(y) - f(x)| < \epsilon$. This means we can pick a $\delta$ ``that works" for all points in $\reals$. Clearly, this definition also extends to general metric spaces.
 
(a) Let f be a continuous real valued function on $[0,\infty)$ such that 
$\lim_{x \ra \infty} f(x)$ exists (and is finite). Prove that $f$
is uniformly continuous on $[0,\infty)$.

(b) Let $f: \reals \to \reals $ be uniformly continuous. If $x_n$ is a Cauchy sequence, show that $f(x_n)$ is also a Cauchy sequence. Is the converse true?

\item Let $(M,d)$ be a metric space. Define a set $\tilde{M} \subset M^\mathbb{N}$ as the collection of all the Cauchy sequences in $(M,d)$.

(a) Show that $\displaystyle{\rho(\{x_n\},\{y_n\}) = \lim_{n \to \infty} d(x_n,y_n)}$ is a well defined function on $\tilde{M} \times \tilde{M}$, i.e the limit always exists.

(b) Show that $\rho$ satisfies the triangle inequality.

(c) Let $a_k = \{x_n\}_k$ be a sequence in $\tilde{M}$ (i.e a sequence of Cauchy sequences in $(M,d)$). Further, assume that $a_k$ is a Cauchy sequence with respect to $\rho$, i.e. for all $\epsilon > 0$, there is an index $K$ such that for all $j,k > K$, we have 
$$
\rho(a_j,a_k) = \lim_{n \to \infty} d((x_n)_j, (x_n)_k) < \epsilon.
$$
Construct an element $a^*$ in $\tilde{M}$ such that $\rho(a_k,a^*) \to 0$ as $k \to \infty$.

\medskip


\end{enumerate}


\section{Compactness}

\begin{enumerate}

\item Let $C([0,1])$ be the set of bounded continuous functions on 
$[0,1]$ with the usual sup norm. So 
\beann
d(f,g) = \sup_{0 \le x \le 1} |f(x)-g(x)|
\eeann
Given a function $g \in C([0,1])$, 
let $U_g=\{f: f(x) > g(x) \quad \forall x \in [0,1]\}$. 
Prove that $U$ is open.  

\item Prove or disprove the following:
\begin{enumerate}
\item $A$ is finite and $U$ is a open subset of $\R$. 
If $A \subseteq U$, there exists an $\epsilon > 0$ such that for 
all $x \in A$, $N(x,\epsilon) \subseteq U$.
\item $P$ is countable and $U$ is a open subset of $\R$. If $P \subseteq U$, 
there exists an $\epsilon > 0$ such that for all $x \in P$, 
$N(x,\epsilon) \subseteq U$.
\item $F$ is closed and $U$ is a open subset of $\R$. 
If $F \subseteq U$, there exists an $\epsilon > 0$ such that for all 
$x \in F$, $N(x,\epsilon) \subseteq U$.
\item $K$ is compact and $U$ is a open subset of $\R$. If $K \subseteq U$, 
there exists an $\epsilon > 0$ such that for all $x \in K$, 
$N(x,\epsilon) \subseteq U$.
\end{enumerate}

\item This is a continuation of problem~2.\ref{prob:boundedmetric}. For $x,y \in \reals$, let 
\beann
d(x,y) = \frac{|x-y|  }{ 1 + |x-y|}
\eeann
We already showed that $d$ is a metric on $\reals$.
\begin{enumerate}
\item Prove that with this metric, the entire space of $\reals$ is not compact
even though it is closed and bounded. 
\item Can you characterize all the compact sets in $(\reals,d)$ ?
\end{enumerate}
  
%\item In a metric space $(X, d)$, a sequence a1, a2, ... of points of 
%X is called a Cauchy sequence
%if for each " > 0 there is a positive integer N such that 
% d(an, am) < " whenever n,m > N.
%A metric space is called complete if every Cauchy sequence in X 
%converges to a point of X. Show that a compact metric space is complete.

\item A metric space $X$ is said to be {\em totally bounded} if for 
any $\epsilon > 0$ it can be covered by finitely many $\epsilon$-balls. 
Also, a metric space is complete, if every Cauchy sequence in $X$ converges. 
Prove that $X$ is compact if and only if $X$ is complete and totally bounded.

\item Define the ``distance'' between two subsets of $\reals$ by $\displaystyle{\rho(A,B) = \inf_{x \in A, y \in B} |x-y|}$.

(a) Is $\rho$ a metric on the power set $2^\reals$ (i.e the collection of all subsets of $\reals$)?

(b) If $A$ is closed and $B$ is compact, show that $\rho(A,B) = 0$ if and only if $A \cap B$ is nonempty.

(c) If $A$ and $B$ are closed, does it follow that $\rho(A,B) = 0 \implies A \cap B \neq \emptyset$?

%\item $l^2$ is the space of square summable sequences of real 
%numbers, i.e., $(x_n)_{n=1}^\infty$
%such that $\sum_{n=1}^\infty x_n^2 < \infty$. The norm is defined by 
%\begin{equation}
%|| (x_n)_{n=1}^\infty|| = \left[ \sum_{n=1}^\infty x_n^2  \right]^{1/2}
%\end{equation}
%\begin{enumerate}
%\item Prove the interior of every compact subset of $l^2$ is empty. 
%\item Define
%\begin{equation}
%F= \{ (x_n)_{n=1}^\infty : \sum_{n=1}^\infty \, n \, x_n^2 \le 1\}
%\end{equation}
%Prove that $F$ is sequentially compact. 
%\end{enumerate}

\end{enumerate}

\section{Calculus}

\begin{enumerate}

\item Prove the inequality $e^x \geq 1+x$ for all $x \in \reals$.

\item $f: \reals^k \to \reals^m$ and $g :\reals^m \to \reals^n$ are differentiable. Prove the chain rule for the composition $g \circ f$. 

\item A $C^2$ function $f : \mathbb{R} \to \mathbb{R}$ is convex if $f''(x) > 0$ for all $x$. Show that, for any finite collection of points $a_1,a_2,\ldots,a_n$, we have the inequality
$$
\frac{f(a_1)+f(a_2) + \cdots + f(a_n)}{n} \geq f\left(\frac{ a_1+a_2+ \cdots+a_n}{n}\right).
$$
(Hint: Draw a picture. Start with $n = 2$)

\item
(a) Give an example of a function of two variables that is discontinuous
at the origin, but whose partial derivatives at the origin exist.

(b) Give an example of a function of two variables all of whose directional
derivatives exist at the origin, but the function itself is not
differentiable at the origin.
\medskip

\item An open subset $O \subset \reals^n$ has the property that for any pair of points $x,y \in O$, there is a differentiable function $\gamma:[0,1] \to O$ such that $\gamma(0) = x$ and $gamma(1) = y$. 
$f : O \ra \reals^m$ is differentiable and $Df = 0$ on $O$. Show that $f$ is a constant on $O$.

(The Hypothesis can be slightly weakened. It is sufficient to assume that $O$ is connected and open.)
\medskip

\item
Evaluate the derivatives of the following matrix functions:

(a) $inv : GL(n) \ra GL(n)$ given by $inv(M) = M^{-1}$.

(b) The determinant function which maps $GL(n)$ to $\reals$. 
\medskip


\item
Show that every point $p$ on the sphere $x^2 + y^2 + z^2 = 1$ 
has a (3 dimensional) neighborhood $U$ such that there is a smooth, one to one
mapping of an open neighborhood $V$ of the origin in $\reals^3$ such that the
plane $z = 0$ maps to the surface of the sphere.
\medskip

\item
Let $f$ and $\frac{\partial f }{ \partial y}$ be continuous on 
$[0, 1] \times [0, 1]$ and assume that $p, q : [0, 1] \ra [0, 1]$ 
are differentiable.
Define
\beann
F(y) = \int_{p(y)}^{q(y)} \, f(x, y) dx, \quad  y \in  [0, 1]
\eeann

Use the chain rule to find $F^\prime(y)$. Hint: Consider 
$G(x_1, x_2, x_3) = \int_{x_1}^{x_2} \, f(t, x_3) dt$.
\medskip

\item
Let $f(x, y) = (x^2 - y^2)/(x^2 + y^2)^2$. Show that
\beann
\int_0^1 \left( \int_0^1 f(x,y) dx \right) dy = -\pi/4, \quad but \quad 
\int_0^1 \left( \int_0^1 f(x,y) dy \right) dx = \pi/4
\eeann
\medskip

\item
Show that in a neighborhood of the origin, the systems of equations
\beann
3x + y - z + u^2 &=& 0 \\
x - y + 2z + u &=& 0 \\
2x + 2y - 3z + 2u &=& 0 \\
\eeann
can be solved for $x, y, u$ in terms of $z$; 
for $x, z, u$ in terms of $y$; 
for $y, z, u$ in terms of $x$.
What can you say (using the implicit function theorem)
about solving them  for
$x, y, z$ in terms of $u$?
\medskip

\item
Let $F$ denote the vector field $(x^2 - z^2, 2xy, z)$ on $\reals^3$. 
Compute in two different ways the surface integral
\beann
\int \int_T \, F \cdot n  \, dA
\eeann
where $T$ denotes the surface of the tetrahedron bounded by 
$x \ge 0, y \ge 0, z \ge 0, x + y + z  \le 1$ 
and $n$ denotes the outward normal to $T$.
Use the two answers to verify the divergence theorem.
\medskip

\item
Compute in two different ways the line integral
\beann
\oint \, ydx - xdy + z^2 dz
\eeann
where $C$ is the intersection of the paraboloid $z = x^2 + 4y^2$ with the
cylinder $x^2 + y^2 = 9$, traversed counter-clockwise when viewed from
the point $(0,0,100)$. Use the two answers to verify Stokes' theorem
\medskip

\item 
A smooth function $F$ on the (open) unit disc in $\reals^2$ satisfies $F(0,0) = (0,0)$ and $\|F(x)\| \geq 3 \|x\|$. 

(a) Show that $F$ is one to one in a neighborhood of the origin. 

(b) Let $F = (u,v)$ in components. Show that, for all $a < 1$, the integral
$$
\frac{1}{2\pi} \oint_{\|r\|=a} \frac{u \nabla v - v \nabla u}{u^2+v^2} \cdot dr
$$
is an integer and this value does not depend on $a$.

(c) Can you determine the value of the integral in terms of the Jacobian $A = DF(0,0)$ of the mapping $F$ at the origin?

(d) Can you construct a smooth function $F$ on the (open) unit disc in $\reals^2$ satisfying $F(0,0) = (0,0)$, $F(x) \neq (0,0)$ for $x \neq (0,0)$, such that the value of the integral is 2? 

\item $M: \mathbb{R}^2 \to \mathbb{R}^2$ is the mapping 
$$
M(x,y) = (2 x - \sin(xy), \frac{1}{3}y + x^2-y^2).
$$

(a) Show that $M$ is invertible in a neighborhood of the origin.

(b) The {\em unstable manifold} of the origin is the set $\Gamma_u = \{(x,y) | M^{-n}(x,y) \to (0,0) \mbox{ as } n \to \infty\}$. Show that $\Gamma_u$ is invariant under $M$.

(c) Show that, there is a unique function $f$ such that $f(x) \to 0$ when $x \to 0$, and the unstable manifold is given by the graph of the function $f$ for sufficiently small $x$, i.e.
$$
\Gamma_u \cap B_{\epsilon}(0,0) = \{(x,f(x)) | |x| < \epsilon\}  \cap B_{\epsilon}(0,0)
$$

\end{enumerate}

\section{Complex analysis}

\begin{enumerate}

\item
Let $f(z)$ be analytic at $c$. Write $f^\prime(c)=r e^{i \theta}$. 
Write $z=x+iy$ and 
$f(z)=u(x,y)+iv(x,y)$. We can think of $f$ as a map $(u(x,y),v(x,y)$ 
from $\reals^2 \ra \reals^2$. The total derivative of this map at $c$ 
is a two by two matrix. Find it in terms of $r$ and $\theta$. 
Express the direction derivatives of the map on $\reals^2$ in terms 
of $r$ and $\theta$. 
\medskip

\item
Define
\beann
f(z) = \sum_{k=1}^\infty a_k (z-z_0)^k
\eeann

\noindent (a) If the radius of convergence of the above power series is $r$, 
then show that the radii of convergence for the series obtained by 
differentiating and integrating the above series termwise is also $r$.

\noindent (b) Show that the termwise differentiated series converges 
uniformly on every disk of the form $|z - z_0 | \le \rho < r$. 
Use this to show that $f(z)$ is given by termwise differentiating the 
above series, for any compact subset of the disk $|z - z_0 | < r$.



\item
Suppose that $\gamma$ is a piecewise smooth positively, 
counterclockwise oriented, simple closed curve. 
Use Green's Theorem to show that the value of the integral
\beann
\oint_\gamma {dz \over z-p} 
\eeann
equals $0$ if $p$ is outside $\gamma$ and $2\pi i$ if $p$ is inside 
$\gamma$.






\item Let $f(z)$ be analytic at $z_0$. Prove that for sufficiently small
$\epsilon$, 
\beann
{1 \over 2 \pi i} \oint_{|z-z_0|=\epsilon} {f(z) \over (z-z_0)^n} \, dz 
= {f^{(n-1)}(c) \over (n-1)!}
\eeann
The contour is the circle centered at $z_0$ with radius $\epsilon$ 
traversed in the counterclockwise direction.
This is a standard theorem in complex analysis books. 
The exercise is to prove it directly from the statement of 
Cauchy's theorem given in the notes. Hint: power series. 

\item 
If $f(z)$ is analytic on the closed disk $B(z_0 ,r)$, 
show that there is a constant $C$ such that 
the derivatives of $f$ at $z_0$ can be bounded by
\beann
\left|f^{(n)}(z_0)\right| \le {C n! \over r^{n+1} }
\eeann
\medskip

\item 
For each of the following functions $f$ and $z_0$, determine 
if the function has a pole or essential singularity at $z_0$. 
In the case of a pole determine the order of the pole and 
the principal part. 

\smallskip
\noindent (a) $f(z)={1 \over z \sin(z)}$, $z_0=0$

\smallskip
\noindent (b) $f(z)={1 \over (z^2+1)^2}$, $z_0=i$

\smallskip
\noindent (c) $f(z)=\exp(-1/z)$, $z_0=0$. 

\smallskip
\noindent (d) $f(z)=\tan^{2}(z)$, $z_0=\pi/2$. 
\medskip

\item
A M\"obius transformation , or a fractional linear transformation, is a
mapping of the form
\beann
w = f (z) = {az + b \over cz + d}
\eeann    
where $a, b, c, d \in \complex$. 
This can be extended to the Riemann sphere = $\complex \cup \{\infty\}$
\beann
f (\infty) = a/c,  \quad   f (-d/c) = \infty
\eeann

\noindent (a) Show that the Mobius transformations form a group.

\noindent   (b) Show that the Mobius transformations map circles to circles on
the Riemann sphere (Note: A straight line on $\complex$ is considered a circle
through $\infty$ on the Riemann sphere).
\medskip


\end{enumerate}

\end{document}

\end


