\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\i{{\mathrm i}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\D{{\mathbb D}}
\def\Re{\operatorname{Re}}
\def\Im{\operatorname{Im}}
\def\interior{\operatorname{int}}

\usepackage{tikz}
\usetikzlibrary{calc,decorations.markings}

\begin{document}

\title{Complex Analysis}
\author{Homework 10}
\date{Due in class Thursday, December 2, 2021}
\maketitle

\begin{enumerate}

\item Show that
\[
  \Gamma(\tfrac12) = \sqrt \pi \,.
\]
\emph{Note:} The proof in Stein \& Sharkarchi goes via Euler's
reflection formula, which is interesting in itself, but lengthy.
There are much shorter proofs that are easy to find.

\item Use Laplace's method to find the leading term in the asymptotic
behavior of
\[
  \int_{-1}^1 \e^{-s \, \cosh x} \, \d x
\]
as $s \to \infty$.

\item Fill in the details of the argument sketched in class that the
Airy function
\[
  \operatorname{Ai}(s) = \frac1{2\pi} \int_{-\infty}^\infty
  \e^{\i (\frac13 x^3+sx)} \, \d x
\]
extends to an entire function on $\C$.

\emph{Hint:} As in class, shift the contour of integration to the line
\[
  L_\delta = \{ x + \i \delta \colon x \in \R \} \,.
\]

\item Show that
\[
  \int_{-\infty}^\infty x \, \e^{\i s x^3} \, \d x
  = \frac\i{\sqrt 3} \, \Gamma(\tfrac23) \, s^{-2/3} \,.
\]

\emph{Hint:} Use contour integration along the contour
\begin{center}
\begin{tikzpicture}
% Configurable parameters
\def\gap{0.2}
\def\bigradius{3}
\def\littleradius{0.4}

% Axes
\draw [-latex] (-1.25*\bigradius, 0) -- (1.25*\bigradius,0);
\draw [-latex] (0, -0.20*\bigradius) -- (0, \bigradius);
% Red path
\draw[line width=1pt]
  (0,0) -- (\bigradius,0) arc (0:60:\bigradius) -- (0,0)
  -- (-\bigradius,0) arc (180:120:\bigradius) -- (0,0);

% The labels
\node at (3.6,-0.3){$\Re$};
\node at (-0.35,3.0) {$\Im$};


\end{tikzpicture}
\end{center}
where the sectors have the opening angle $\pi/3$.

\end{enumerate}



\end{document}