\documentclass[11pt]{article}

  \usepackage{latexsym}
  \usepackage{comment}
  \usepackage{showlabels}
  \usepackage{amsmath,amsthm,amsfonts,amssymb,graphicx,epsfig,latexsym,color,amssymb}
  %\usepackage{showkeys}
  \usepackage[shortlabels]{enumitem}
  \def\rank{\operatorname{rank}}

  \newtheorem{thm}{Theorem}%[section]
  \newtheorem{lemma}[thm]{Lemma}
  \newtheorem{claim}[thm]{Claim}
  \newtheorem{cor}[thm]{Corollary}
  \newtheorem{prop}[thm]{Proposition}
  \newtheorem*{main}{Main Theorem}{}
  \newtheorem*{rrt}{Rank Rigidity Theorem}{}

  \theoremstyle{remark}
  \newtheorem{example}[thm]{Example}
  \newtheorem{exercise}[thm]{Exercise}
  \newtheorem{definition}[thm]{Definition}
  \newtheorem{remark}[thm]{Remark}
  \newtheorem{examples}[thm]{Examples}
  \newtheorem*{definition*}{Definition}
  \newtheorem*{remark*}{Remark}
  \newtheorem{question}[thm]{Question}

  \newcommand{\mb}[1]{\marginpar{\tiny #1 --mb}}

  \bibliographystyle{plain}   %plain

  \def\R{{\mathbb R}}
  \def\Q{{\mathbb Q}}

  \def\Z{{\mathbb Z}}
  \def\N{{\mathbb N}}
  \def\H{{\mathbb H}}  
  \def\G{{\Gamma}}
  \def\S{{\mathcal S}}
  \def\s{{\sigma}}
  \def\t{{\tau}}
  \def\e{{\varepsilon}}
  \def\F{{\mathcal F}}
  \def\X{{\mathcal X}}
  \def\P{{\mathcal P}}
  \def\pa{pseudo-Anosov\ }
  \def\diam{\operatorname{diam}}
  \def\spec{\operatorname{spec}}
  \def\stab{\operatorname{Stab}}
  \def\diag{\operatorname{diag}}
  \def\SL{\operatorname{SL}}
  \def\GL{\operatorname{GL}}
  \def\Sym{\operatorname{Sym}}
  \def\FF{{\mathbb F}}
  \def\A{{\mathcal A}}
  \def\l{{\lambda}}
  \def\C{{\mathbb C}}

  \newcommand{\<}{\langle}
  \renewcommand{\>}{\rangle}

\begin{document}

\centerline{\Large Homework 5: Singularities, Residue Calculus}
\vskip 1cm

{\bf Singularities}

\begin{enumerate}[1.]

\item If $f$ is a holomorphic function defined in $\{|z|>R\}$ (we
  think of this set as a neighborhood of $\infty$) we say that
  $\infty$ is a removable or essential singularity or a pole provided
  that 0 is the respective singularity for the function $g(z)=f(\frac
  1z)$. Show that
  \begin{enumerate}[(i)]
  \item A nonconstant polynomial has a pole at infinity.
    \item If $f$ is an entire function which is not a polynomial, then
      $f$ has an essential singularity at $\infty$.
  \end{enumerate}
  \end{enumerate}

{\bf Residue Calculus.}
\begin{enumerate}[resume]
\item Let $P$ be a polynomial of degree $\geq 2$.
  \begin{enumerate}[(a)]
    \item Show that for any
  circle $C$ of big enough radius so that it encloses all roots we
  have
  $$\int_C \frac {dz}{P(z)}=0$$
  \item Assuming
  all roots
          $z_1,\cdots,z_n$ of $P$ are distinct prove (using the
  Residue theorem) that
  $$\sum_{j=1}^n \frac 1{P'(z_j)}=0$$
  \end{enumerate}
  
\item Compute
  $$\frac 1{2\pi i}\int_{|z|=1}\sin\bigg(\frac 1z\bigg) dz$$
  
\item Show that
  $$\int_0^{2\pi} \frac {d\theta}{1-2r \cos\theta+r^2}=\frac
  {2\pi}{|1-r^2|}$$ when $r\in\R\smallsetminus\{-1,1\}$.

\item Prove the Wallis formula
  $$\frac 1{2\pi} \int_0^{2\pi} (2\cos\theta)^{2m}d\theta=\binom
  {2m}m$$ for $m=1,2,\cdots$.


\item Prove that
  $$\int_0^\infty \frac {dx}{1+x^n} =
  \frac{\frac{\pi}n}{\sin\big(\frac{\pi}n\big)} $$ for $n=2,3,\cdots$.

  Hint: Use the contour consisting of $[0,R], [0,Re^{2\pi i/n}]$ and
  the short arc of $|z|=R$ connecting the endpoints.

\item Prove Jordan's lemma: Let $F$ be a continuous function defined
  in $\{z\in\C\mid |z|\geq R_0, Im(z)\geq 0\}$ satisfying $F(z)\to 0$
  when $z\to\infty$. If $m>0$ and $\Gamma_R$ denotes the semicircle
  $\{z\mid |z|=R, Im(z)\geq 0\}$, then
  $$\int_{\Gamma_R}e^{imz}F(z)~ dz\to 0\qquad\mbox{ as } R\to\infty$$
  Hint: When the dust settles you will want to show that
  $R\int_{0}^{\pi}e^{-mR sin(t)}dt$ is uniformly bounded as
  $R\to\infty$. It is convenient to break the integral into
  $\int_0^{\pi/2}$ and $\int_{\pi/2}^\pi$. E.g. on the first one argue
  that for some $\epsilon>0$ we have $sin(t)\geq \epsilon t$ on
  $[0,\pi/2]$ and then you can calculate the integral with $\epsilon t$ in
  place of $sin(t)$.

\item Prove that
  $$\int_{-\infty}^\infty \frac {x^3\sin x}{(x^2+1)^2} dx=\frac{\pi}{2e}$$
  Hint: As usual, $f(z)=\frac {z^3e^{iz}}{(z^2+1)^2}$.

\item Prove that
  $$\int_0^\infty \frac{x\sin x}{x^4+1} dx=\frac{\pi}2 e^{-1/\sqrt
  2}\sin\bigg( \frac 1{\sqrt 2}\bigg)$$

\item Prove that
  $$\int_{-\infty}^\infty \frac {\cos x}{\cosh x} dx = \frac
  {\pi}{\cosh\big(\frac {\pi}2\big)}$$

  Hint: For the contour take the rectangle of height $\pi$ and base
  $[-R,R]$.


\item Prove that
  $$\sum_{n=1}^\infty \frac 1{n^4}=\frac {\pi^4}{90}$$
  This can be done in many ways, including Fourier series, but here
  you should use Residue Calculus.

\item Prove that
  $$1-\frac 1{2^2}+\frac 1{3^2}-\frac 1{4^2}+\frac 1{5^2}-\cdots=\frac
  {\pi^2}{12}$$
    Hint: $\pi\csc \pi z=\frac {\pi}{\sin \pi z}$ has residue $(-1)^n$ at $z=n$.

\item We showed in class that if $\phi:C\to\C$ is a continuous
  function defined on the circle $|z|=1$ then the Cauchy formula
  $$f(z)=\frac 1{2\pi i}\int_C \frac{\phi(\xi)}{\xi-z}~ d\xi$$
  defines a holomorphic function in $\mathbb D=\{|z|<1\}$.
  \begin{enumerate}[(i)]
  \item Compute $f$ when $\phi(\xi)=\overline\xi=1/\xi$.
  \item Compute $f$ when $\phi(\xi)=\xi^n$ for $n\in\Z$.
  \item Compute $f$ when $\phi$ is given by its Fourier series
    $$\phi(\xi)=\sum_{n\in\Z} a_n\xi^n$$
    under the assumption that convergence of the series is uniform.
  \end{enumerate}
  Hint: The answer to (iii) is to remove the negative powers. So from
  this point of view holomorphic functions form a ``half-dimensional''
  subspace of the space of functions on the circle.
    

\end{enumerate}


  

\end{document}
