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\title{Complex Analysis HW II}



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\maketitle

   This homework is due by class time on Monday 24 Feb. There is a long preamble before the questions start, you don't have to prove anything
   stated in the preamble.

\medskip

   Recall the discussion from class of the ``extended complex plane'' and  ``Riemann sphere''. We took the sphere $S$ in ${\mathbb R}^3$ given by
   the equation $x^2 + y^2 + z^2 = 1$, defined a bijection between $S$ and ${\mathbb C} \cup \infty$ (in  which the
   ``north pole'' $(0, 0, 1)$ corresponds to $\infty$ and the ``south pole'' $(0, 0, -1)$ corresponds to $0$)
   and then used this bijection to define a metric on
   ${\mathbb C} \cup \infty$ by copying the Euclidean metric on $S$. This metric is different from the usual one but
   a set $U \subseteq {\mathbb C}$ is open in the usual metric iff it is open in this new metric.
   
   We would like to do ``complex analysis on ${\mathbb C} \cup \infty$'', that is to consider functions 
   whose domain or codomain may include the point $\infty$. To do this, as was briefly discussed in class,
   we will introduce two coordinate systems which will parametrise open subsets $U_1$ and $U_2$ of  ${\mathbb C} \cup \infty$
   using coordinates from $\mathbb C$. 

\begin{enumerate}

\item  $U_1 = {\mathbb C}$, $f_1: {\mathbb C} \rightarrow U_1$, $f_1(z) = z$. 

\item  $U_2 = {\mathbb C} \cup \{ \infty \} \setminus \{ 0 \}$, $f_2: {\mathbb C} \rightarrow U_2$, $f_2(z) = 1/z$
     with the obvious convention that $1/0 = \infty$. 

\end{enumerate} 
     It is easy to see that $f_1$ and $f_2$ are continuous bijections with continuous inverses. In terms of the Riemann
   sphere, $f_1$ sets up a bijection between $\mathbb C$ and the sphere with the north pole deleted, 
    while $f_2$ sets up a bijection between $\mathbb C$ and the sphere with the south pole deleted. 


   Observation: $U_1 \cap U_2 = {\mathbb C} \setminus \{ 0 \}$. A point $z \in U_1 \cap U_2$ has coordinate $z$ with respect to $f_1$ and
   coordinate $1/z$ with respect to $U_2$. The function $z \mapsto 1/z$ is holomorphic and has a holomorphic inverse
   on ${\mathbb C} \setminus \{ 0 \}$.


   Let $f$ be a function from an open subset $\Omega$ of  ${\mathbb C} \cup \{ \infty \}$ to ${\mathbb C} \cup \{ \infty \}$, we will
   say what it means for $f$ to be holomorphic. Let $a \in \Omega$, let $f(a) = b$ and find $i$ and $j$ such that $a \in U_i$ and
   $b \in U_j$. The function $f_j^{-1}  \circ f \circ f_i$ is defined (and takes values in $\mathbb C$) in a open neighbourhood 
   of $f_i^{-1}(a)$, we will say that $f$ is holomorphic at $a$ iff $f_j^{-1}  \circ f \circ f_i$ is 
   holomorphic at $f_i^{-1}(a)$.
   $f$ is holomorphic iff it is holomorphic at every $a \in \Omega$.  

  Key point: Although the choice of $i$ and $j$ may not be unique, there is no ambiguity in the definition because 
  in the overlap between $U_1$ and $U_2$ the function which translates between the two coordinate systems
  is holomorphic and has a holomorphic inverse.

  Easy example: The function $z \mapsto z^2$ (with the convention that $\infty^2 = \infty$) is a 
  holomorphic function from ${\mathbb C} \cup \infty$ to itself.     


\begin{enumerate} 

\item   Prove that if $f$ is a holomorphic function from $\mathbb C$ to $\mathbb C$, and
    there is a constant $C > 0$ and a natural number $k$ such that $\vert f(z) \vert \le C \vert z \vert^k$
    for all $z$ with $\vert z \vert > 1$, then $f$ is a polynomial of degree at most $k$.
    Hint: estimate $f^{(k+1)}$ using the Integral Formula. 



\item  Let $f$ be holomorphic on a region $\Omega \subseteq {\mathbb C}$, and suppose that 
      $a \notin \Omega$ but there is $\delta > 0$ such that $B(a, \delta) \setminus \{ a \} \subseteq \Omega$.

Prove that the following are equivalent:

\begin{enumerate}
\item $f$ has a pole at $a$.
\item The function $f_1$ from $\Omega \cup \{ a \}$ to $\mathbb C \cup \{ \infty \}$
 obtained by setting   $f_1 \restriction \Omega = f$, $f_1(a) = \infty$ is holomorphic. 
\end{enumerate}



\item   Let $a, b \in {\mathbb C}$. Prove that the function $z \mapsto  a z  + b$ is a holomorphic function
  from ${\mathbb C} \cup \{ \infty \}$ to ${\mathbb C} \cup \{ \infty \}$, where we make the obvious convention that
  $a \times \infty + b = \infty$ for $a \neq 0$, $0 \times \infty + b = b$. Prove that $1/z$ is also such a function, with the conventions that
  $1/0 = \infty$ and $1/\infty = 0$. 

\item  Say that a bijection $f$ from ${\mathbb C} \cup \{ \infty \}$ to itself is an {\em automorphism}
    if $f$ is holomorphic and has a holomorphic inverse. We already saw that $\phi: z \mapsto 1/z$ and
    $\psi_{a, b} : z \mapsto a z + b$ for $a \neq 0$ are automorphisms. Describe the group of automorphisms
    generated by $\{ \phi \} \cup \{ \psi_{a, b}: a, b \in {\mathbb C}, a \neq 0 \}$. 



\item  Prove that a holomorphic function from ${\mathbb C} \cup \{ \infty \}$ to $\mathbb C$ is constant.
   Hint:   The Riemann sphere is compact. 


\item  Let $f$ be a non-constant holomorphic function from ${\mathbb C} \cup \{ \infty \}$ to ${\mathbb C} \cup \{ \infty \}$.
    Prove that $\{ a : f(a) = 0 \}$ and $\{ a : f(a) = \infty \}$ are finite sets. 
  Hint:   The Riemann sphere is compact. 

\item  Prove that if $f$ is a non-constant holomorphic function from ${\mathbb C} \cup \{ \infty \}$ to ${\mathbb C} \cup \{ \infty \}$,
    then there is an automorphism $\Phi$ such that $f \circ \Phi$ has neither a zero nor a pole at $\infty$. 
Hint: Use the last few questions. 
 
\item  Prove that if $f$ is a non-constant holomorphic function from ${\mathbb C} \cup \{ \infty \}$ to ${\mathbb C} \cup \{ \infty \}$
    then $f$ has an equal number of zeroes and poles (counted up to multiplicity).
   Hint: Use the previous question to move all the zeroes and poles into $\mathbb C$, and draw a big circular contour
   $\gamma$ which encloses them all.

  







\end{enumerate}

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