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\begin{document}

\centerline{\Large Homework 2: Integration along paths}
\vskip 1cm

This homework set is more along the lines of ``routine calculations'',
but the goal is to get comfortable doing these kinds of calculations
as there will be many of them throughout the semester.

Recall that a {\it primitive} for a function $f$ is a function $F$
such that $F'=f$. In this situation, the Fundamental Theorem of
Calculus states that $\int_\gamma f(z)dz=F(w_2)-F(w_1)$ if $\gamma$ is
a piecewise smooth path from $w_1$ to $w_2$.

Several problems here use the Estimation Theorem:
$$\bigg|\int_\gamma f(z)dz\bigg|\leq L(\gamma) \max_{x\in\gamma} |f(z)|$$
where $L(\gamma)$ is the length of $\gamma$.


\begin{enumerate}[1.]

\item Evaluate the following integrals $\int_\gamma f(z)dz$. Where
  possible, use the existence of primitives and the Fundamental
  Theorem of Calculus. Problem (iii) can be reduced to the Fundamental
  Calculation we did in class.
  \begin{enumerate}[(i)]
  \item $f(z)=z^2$, $\gamma(t)=e^{it}$, $t\in [-\pi/2,\pi/2]$.
    \item $f(z)=e^z$, $\gamma$ is the join of the segments $[0,1]$,
      $[1,1+i]$ and $[1+i,i]$.
    \item $f(z)=1/z$, $\gamma(t)=e^{-it}$, $t\in [0,8\pi]$.
  \end{enumerate}

\item Evaluate the following integrals $\int_\gamma f(z)dz$.
  \begin{enumerate}[(i)]
  \item $f(z)=Re(z)$, $\gamma(t)=t+it^2$, $t\in [0,1]$.
  \item $f(z)=|z|^4$, $\gamma$ is the interval $[-1+i,1+i]$.
  \end{enumerate}

  \item Evaluate the following integrals $\int_\gamma f(z)dz$ where
    $\gamma=\gamma(0,1)$ is the circle centered at 0 with radius 1 oriented
    counterclockwise. Where possible, use the Fundamental Calculation
    from class or the Fundamental Theorem of Calculus. For (i) recall
    that the integral depends only on the values of $f$ along
    $\gamma$, and maybe you can find a better function $g$ such that
    $g=f$ on $\gamma$.
    \begin{enumerate}[(i)]
    \item $f(z)=|z|^4$.
    \item $f(z)=z^{-2}(z^4-1)$.
    \item $f(z)=\sin z$.
    \end{enumerate}

    \item Use the Estimation Theorem to prove the following upper
      bounds. We denote by $\gamma(c,r)$ the circle of radius $r$
      centered at $c$.
      \begin{enumerate}[(i)]
      \item $|\int_{\gamma(1,2)} \frac 1z dz|\leq 4\pi$.
        \item $|\int_{\gamma(0,r)} \frac{z-1}{z+1} dz|\leq \frac {2\pi
          r(r+1)}{|r-1|}$.
          \item $|\int_{\Gamma_R} \frac{e^{iz}}{z^4}dz|\leq \pi
            R^{-3}$, where $\Gamma_R$ is the semicircle
            $$\Gamma_R=\gamma(0,R)\cap \{z\mid Im(z)\geq 0\}$$
          \item $|\int_{[0,1+i]}(z^2+1)^{-1}dz|\leq\sqrt 2$.
      \end{enumerate}
      
  

  \item 
    \begin{enumerate}[(a)]
    \item Let $p(z)$ be a polynomial. Show that for any closed path
      $\gamma$ in $\C$
      $$\int_\gamma p(z) dz=0$$
    \item Show that for some $\epsilon>0$ the following holds. If
      $p(z)$ is any polynomial then there is some $z_0$ with $|z_0|=1$
      such that $|p(z_0)-1/z_0|\geq \epsilon$. Thus the function $1/z$
      cannot be uniformly approximated by polynomials on the unit
      circle. Find the best possible $\epsilon$ and for that
      $\epsilon$ find an approximating polynomial.
    \end{enumerate}
\end{enumerate}


\subsection*{Cauchy integral formula}

In the following problems, $f(z)=\frac{e^{z^2}}{z-2}$. This is
holomorphic on $\C\smallsetminus\{2\}$. You should use Cauchy's
theorem or
integral formula to compute the following integrals. $\gamma(a,r)$
denotes the circle centered at $a$ with radius $r$, oriented
counterclockwise. 

\begin{enumerate}[resume]
\item $\int_{\gamma(2+i,2)}f(z)~ dz$
\item $\int_{\gamma(3+i,1)}f(z)~ dz$
\item $\int_{\alpha}f(z)~ dz$, where $\alpha$ is the curve shown.
\end{enumerate}
\begin{center}
  %\begin{figure}[h]
    \includegraphics[scale=0.3]{alpha.pdf}
    %\end{figure}
  \end{center}


    \subsection*{Cauchy inequalities}
    \begin{enumerate}[resume]

\item Suppose $f$ is an entire function (holomorphic on all of $\C$)
  that satisfies $|f(z)|\leq A|z|^n+B$ for some integer $n\geq 0$ and $A,B>0$. Show that $f$
  is a polynomial.

    \end{enumerate}

    \subsection*{Power series}
Just so we don't forget power series, here is another problem
involving them.
\begin{enumerate}[resume]
    
\item
Prove that for $|z|<1$ the following identity holds. Justify any
manipulations with power series.
$$\frac z{1-z^2}+\frac {z^2}{1-z^4}+\frac {z^4}{1-z^8}+\cdots+\frac
{z^{2^n}}{1-z^{2^{n+1}}}+\cdots=\frac z{1-z}$$

  
      \end{enumerate}
  

\end{document}
