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\begin{document}

\centerline{\Large Homework 1: Cauchy-Riemann equations, power series}
\vskip 1cm

Let $\Omega\subset\C=\R^2$ be a domain (connected open set).

\subsection*{Holomorphic functions.}



\begin{enumerate}[1.]

    \item \label{nonzero derivative}
      Suppose $f:\Omega\to\C$ is holomorphic. Suppose that
     $f'(z_0)\neq 0$ for some $z_0\in\Omega$. Show that for any
     $v\in\C$ with $|v|=1$ there is a sequence $w_n$ in the image of
     $f$ converging to $f(z_0)$ so that $w_n\neq f(z_0)$ and such that
     $\frac{w_n-f(z_0)}{|w_n-f(z_0)|}\to v$. Thus $w_n$ converges to
      $f(z_0)$ from the prescribed direction. By a change of
      coordinates you may assume that $z_0=f(z_0)=0$. Note: Later we
      will see that $f(z_0)$ is contained in the interior of the image
      of $f$.

     \item Use Problem \ref{nonzero derivative} to deduce that if the image
     of $f$ is contained in a smooth curve, e.g. a line or a circle, then $f$ is
     constant.
\end{enumerate}

\subsection*{Cauchy-Riemann equations.}

\begin{enumerate}[resume]

  

  \item Suppose $\phi,\psi:\Omega\to\R$ are differentiable and satisfy
    Cauchy-Riemann equations $$\frac{\partial\phi}{\partial
      x}=\frac{\partial\psi}{\partial y}, \qquad \frac{\partial\phi}{\partial
      y}=-\frac{\partial\psi}{\partial x}$$
    Show that pairs of functions
    \begin{enumerate}[(a)]
    \item $\phi_1=\phi^2-\psi^2$, $\psi_1=2\phi\psi$,
    \item $\phi_2=e^\phi \cos\psi$, $\psi_2=e^\phi \sin\psi$,
    \item $\phi_3=-\psi$, $\psi_3=\phi$
    \end{enumerate}
    also
      satisfy the Cauchy-Riemann equations. In each case represent the
      holomorphic function $f_j=\phi_j+i \psi_j$ directly in terms of
      $f=\phi+ i \psi$.


     
     

\item A function $u:\Omega\to\R$ is called {\it
  harmonic}\footnote{strictly speaking, being harmonic is a local
  property and the definition I am giving is correct only when
  $\Omega$ is simply connected} if $u=Re(f)$
  for some holomorphic function $f:\Omega\to\C$.
  \begin{enumerate}[(a)]
  \item Show that $Im(f)$ is also harmonic when $f$ is holomorphic.
    \item If $u:\Omega\to\R$ is harmonic, up
      to adding a constant there is a unique $v:\Omega\to\R$ such that
      $u+iv$ is holomorphic.
      \item Suppose that $u:\Omega\to\R$ is harmonic and has
        continuous second
        partial derivatives (this is always true as we shall see
        later). Show that $u$ satisfies the Laplace equation
        $$\Delta u:=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2
          u}{\partial y^2}=0$$
  \end{enumerate}
\end{enumerate}


  \subsection*{Power series.}
  \begin{enumerate}[resume]
  \item \label{eq1} Prove that $e^{z+w}=e^ze^w$ for any $z,w\in\C$.
  \item \label{eq2} Prove that $sin^2z+cos^2z=1$ for any $z\in\C$.
    \item \label{eq3} Prove that $\sin(z+w)=\sin(z)\cos(w)+\cos(w)\sin(z)$,
      $z,w\in\C$.
      \item \label{eq4} Prove that $\cos(z+w)=\cos(z)\cos(w)-\sin(z)\sin(w)$,
        $z,w\in\C$. 
    \item Show that $\sum_n nz^n$ has radius of convergence 1 but does
      not converge for {\it any} $z$ on the unit circle.
      \item Show that $\sum_n z^n/n^2$ has radius of convergence 1 and
      converges for {\it every} $z$ on the unit circle (in fact,
      absolutely and uniformly on $\{|z|\leq 1\}$).
    \item An {\it arithmetic sequence} is a sequence of integers of
      the form $a,a+d,a+2d,\cdots$ with $d>0$ called the {\it
        step}. Show that $\mathbb N=\{1,2,3,\cdots\}$ cannot be
      partitioned into finitely many subsets each of which forms an
      arithmetic sequence with pairwise distinct steps. Hint:
      Otherwise write
      $1/(1-z)$ as the sum of finitely many functions of the form
      $z^a/(1-z^d)$. 

  \end{enumerate}

  Comment: In Problems \ref{eq1}-\ref{eq4} the intent is to manipulate the
  defining power series, using the fact that they are absolutely
  convergent. In the next few lectures an easier argument will emerge,
  based on the fact that if two holomorphic functions defined on $\C$
  agree on $\R$ then they agree everywhere. Then Problem \ref{eq2} is immediate, and
  for the others you can do it in two steps. E.g. for Problem
  \ref{eq1} first fix $w\in\R$ and argue
  that $z\mapsto e^{z+w}$ and $z\mapsto e^ze^w$ are equal. Then repeat
  this with $w\in\C$. There are other tricks too, e.g. the trig
  formulas follow from Problem \ref{eq1} and the Euler formula.

  \subsection*{Abel's theorem and convergence on the circle $|z|=R$.}
  
  The goal of the next exercise is to show that it is legitimate to plug
  in $z=1$ in the power series
  \begin{itemize}
  \item $\log(1+z)=z-z^2/2+z^3/3-\cdots$
  \item $\arctan z=z-z^3/3+z^5/5-\cdots$
  \end{itemize}
  even though the radius of convergence is 1,
  thus proving that $$1-\frac 12+\frac 13-\cdots=\log 2$$ and
  $$1-\frac 13+\frac 15-\cdots=\pi/4$$
  The key property is that the series happens to converge to {\it something}
  for $z=1$.
  \begin{enumerate}[resume]
    \item
    \begin{enumerate}[(a)]
      \item Let $a_0,a_1,\cdots$ and $b_0,b_1,\cdots$ be two sequences
        of complex numbers and set $s_n=a_0+a_1+\cdots+a_n$. Prove 
        {\it Abel's summation by parts} for $n\geq 0$, $p\geq 1$:
        $$\sum_{k=n+1}^{n+p} a_kb_k=\sum_{k=n+1}^{n+p}
        s_k(b_k-b_{k+1})-s_nb_{n+1}+s_{n+p}b_{n+p+1}$$
        Notice the similarity with $\int udv=uv-\int vdu$.
      \item Now let $\sum a_nz^n$ be a power series with radius of
        convergence 1 and assume that $\sum a_n$ converges. Abel's
        theorem states that convergence of $\sum a_nz^n$ is uniform
        (though not necessarily absolute) on any closed sector centered at 1
        and contained in $\{|z|<1\}\cup\{1\}$. In
        particular, the limiting function is continuous on the
        sector. This
        justifies the validity of plugging in $z=1$.
        Pictured
        is a sector with bounding lines of slope $\pm 1$.
        
        \begin{figure}[h]
          \begin{center}
            \includegraphics[scale=0.3]{abel.pdf}
            \end{center}
        \end{figure}
        
        
        Prove
        Abel's theorem. It might
        make it easier to assume the sector is just $[0,1]$, which is
        sufficient for the examples.


        Hint: First show that there is a constant $M>0$ so that
        $|1-z|\leq M (1-|z|)$ on the sector (e.g. $M=1$ for the
        degenerate sector $[0,1]$). To simplify the situation, add a
        constant so that $\sum a_n=0$. Take $b_n=z^n$ in Abel's
        summation. For any $\epsilon>0$ choose $n$ so that
        $|s_k|<\epsilon$ for $k\geq n$. Then estimate the main sum by
        $\epsilon |1-z| |z|^n \frac 1{1-|z|}$.
        
    \end{enumerate}

    \item This might be called the anti-Abel's theorem. Suppose
      $a_0\geq a_1\geq a_2\geq\cdots\to 0$ and assume $\sum a_nz^n$
      has radius of convergence 1. Example: $a_n=1/n$. Thus the series
      may not converge for $z=1$. However, show that it converges for
      any other $z$ with $|z|=1$. In fact, show that the convergence
      is uniform on any set of the form $\{z\mid |z|\leq 1,
      |1-z|\geq\delta\}$ for any $\delta>0$. Hint: Abel's summation
      again, this time set it up so that the $a_k$ from the summation
      is $z^k$ and $b_k$ is $a_k$ from the series (sorry about the
      notational conflict).

    \item Let $z_1,z_2,\cdots,z_p$ be distinct points on the unit circle.
      Show that the following power series with radius of
        convergence 1 diverges for $z=z_1,z_2,\cdots,z_p$ but
        converges at all other points on the unit circle:
        $$\sum_{n=1}^\infty \frac 1n \bigg(\frac 1{z_1^n}+\frac
        1{z_2^n}+\cdots+\frac 1{z_p^n}\bigg)z^n$$
  \end{enumerate}
  

\end{document}
