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\title{Math 122B: Take home final}
\date{}


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\begin{document}
\maketitle

Any theorems covered during lecture may be referenced.  Any other theorems or nontrivial claims must be provided with proof. A claim is nontrivial if you do not know why its true., i.e. no ``From Theorem 2 of Book X''.

\textbf{Due in class on Thursday or online Friday}

\begin{enumerate}
\item Show that $\displaystyle \int_{S_N} \frac{dz}{z^3\cos(\pi z)} \to 0$ as $N\to \infty$, where $S_N$ is the square $[-N,N]\times [-iN,iN]$.  Use this to compute $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3} = \frac{\pi^3}{32}$.

\item Let $f$ be holomorphic on $\mathbb{C}$ and define the function
\begin{equation*}
M(f,r) = \frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2d\theta.
\end{equation*}
Show that $M$ is strictly increasing in $r$, unless $f$ is a constant.  Hint: Consider the series expansion of $f$ centered at $0$.

Using the result, find all entire (holomorphic on $\mathbb{C}$) functions such that
\begin{equation*}
\iint_{\mathbb{C}}|f(z)|^2dA < \infty
\end{equation*}


\item Compute the integral $\displaystyle \int_0^\infty \frac{dx}{x^a+1}$  
for any real $a >1$

\item Let $F$ be holomorphic in $|z| < 1+\eps$ for some $\eps > 0$, and maps the closed unit disk into the open unit disk.  Show that $F$ has a fixed point, i.e. there is a $z$ in the open unit disk such that $F(z) = z$.  Hint: Rouch\'e's theorem with $f(z) = -2z$ and $g(z) = F(z) +z$.

\item Assume that $f$ is holomorphic and one-to-one in $D$, hence an inverse $f^{-1}$ exists.  Show that
\begin{equation*}
g(z) = \frac{1}{2\pi i}\int_{\gamma}\frac{sf'(s)}{f(s)-z}ds
\end{equation*}
where $\gamma$ is a closed simple contour, defines the inverse of $f$ inside $\gamma$,i.e., show that $g(f(z_0)) = z_0$.  In particular, this would show that $f^{-1}$ is holomorphic.  Note we cannot use the Cauchy integral formula on $f^{-1}$ directly since we have not shown that $f^{-1}$ is holomorphic yet, hence you would have to show that fact in another method before applying Cauchy integral formula.  Consider multiplying and dividing by $z-z_0$.  

\item Show that any open disk is conformally equivalent to any open half-plane.  

\item Let $n$ be a positive integer.  Prove that $\int_0^{2\pi}e^{\cos\theta}\cos(n\theta-\sin\theta)d\theta = \frac{2\pi}{n!}$

\end{enumerate}

\end{document}