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\begin{document}

\centerline{\Large Homework 3: Cauchy, Morera, Integrals, Runge}
\vskip 1cm

\subsection*{Cauchy inequalities, Liouville}

\begin{enumerate}[1.]

  \item Suppose that $f$ and $g$ are entire functions such that
    $|f(z)|\leq |g(z)|$ for all $z\in\C$. Show that there is a complex
    number $\lambda$ such that $f(z)=\lambda g(z)$ for all
    $z\in\C$. Warning: If you consider $f/g$ you should argue that it
    is well-defined at the zeros of $g$.


      \item Let $f(z)=\sum_{n=0}^\infty a_n z^n$ be a power series
        with radius of convergence $\geq 1$. Suppose that $|f'(z)|\leq 1$ for
        all $z$ with $|z|<1$. Prove that $|a_n|\leq \frac 1n$ for all
        $n$. By examples, show that these inequalities are sharp. Hint:
        Consider $g=f'$ and apply Cauchy inequalities on the circle of
        radius $1-\epsilon$ and then let $\epsilon\to 0$. The examples
        are one for each $n$ separately, not one for all $n$.
\end{enumerate}

\subsection*{Morera's theorem.}
\begin{enumerate}[resume]

        \item Prove the following version of Morera's theorem. Suppose
          $f:\Omega\to\C$ is continuous and $\Omega$ is the open
          rectange $\{z\in\C\mid Re(z)\in (-1,1), Im(z)\in
          (-1,1)\}$. Suppose that $\int_\gamma f(z)dz=0$ for every
          rectangle $\gamma$ in $\Omega$ with sides parallel to the
          real and imaginary axes. Show that $f$ is holomorphic. Note:
          The assumption that $\Omega$ is a rectangle is just for
          convenience; the statement is true for any open set. Hint:
          Construct a primitive just like in the triangle version.
\end{enumerate}



\subsection*{Integrals.}
In the first three problems use the same curve we used in class,
consisting of segments $[-R,-\epsilon]$, $[\epsilon,R]$ and the two
semicircles centered at 0 of radius $\epsilon$ and $R$. Recall that
$\int_{0}^\infty$ by definition is $\lim_{\epsilon\to
  0+,R\to\infty}\int_{\epsilon}^R$. 

\begin{enumerate}[resume]
            
\item (Dirichlet integral)\label{Dirichlet}
  $\int_0^\infty \frac {\sin x}x dx=\pi/2$. Hint:
  $e^{iz}/z$. There is an additional trick here. You will have to
  subdivide the semicircle between $R$ and $-R$ into 3 subarcs (two
  small ones near $R$ and $-R$) and
  apply the Estimation Theorem separately on them.
\item $\int_{0}^{\infty} \big(\frac {\sin x}x\big)^2dx=\pi/2$. Hint:
  $\frac{1-e^{2iz}}{z^2}$. Actually, this integral is equivalent to
  the one we did in class after a simple substitution.
  \item Prove that $\int_{0}^{\infty} \big(\frac {\sin
    x}x\big)^3dx=\frac{3\pi}8$. Hint: $\frac{3e^{iz}-e^{3iz}}{z^3}$.

  \item Compute the Fresnel integrals
    $$\int_{0}^{\infty} \cos(t^2)dt=\int_{0}^{\infty} \sin(t^2)dt=\sqrt{\frac{\pi}8}$$
    You can use the Gaussian integral
    calculation $$\int_{-\infty}^{\infty} e^{-t^2}dt=\sqrt{\pi}$$ from
    real analysis. Hint: Consider $f(z)=e^{-z^2}$ and integrate on the
    sector that consists of segments $[0,R]$, $[0,Re^{i\pi/4}]$ and
    the circular arc connecting $R$ and $Re^{i\pi/4}$. As in Problem
    \ref{Dirichlet} you will have to subdivide the circular arc.
  
      \end{enumerate}

\subsection*{Runge's theorem}

\begin{enumerate}[resume]

  \item In class we constructed a sequence of polynomials that pointwise
  converges to a discontinuous function. Find a variant of this
  construction to show that there is a sequence of polynomials that
  pointwise converges to the zero function on $\C$, but not uniformly
  in any neighborhood of 0. Comment: We will see later that if a
  sequence of holomorphic functions converges pointwise to $f$ and all
  the functions are uniformly bounded on every compact set, then the
  convergence is uniform on compact sets and $f$ is holomorphic.
\end{enumerate}



\end{document}
