\documentclass[12pt]{amsart}
\title[Problem Sheet 8]{Complex Analysis: Problem Sheet 8}
\author{}
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\begin{document}
\maketitle

\noindent {\bf I. Problems from Lecture 28.}
Problems 1--3 below use Jordan's lemma from \S 28.1 and Lemma 28.4 to apply
the technique of indenting a contour (see Figure \ref{fig:indent} below).
\begin{figure}[h]
\includegraphics[width=3in]{indent.pdf}
\caption{Indented contour $C_{r,R}$ consisting
of $4$ smooth pieces.}
\label{fig:indent}
\end{figure}

\problem{1} Prove that $\ds\int_0^{\infty} \frac{\sin(x)}{x}\, dx
= \frac{\pi}{2}$.\\

\problem{2} Prove that $\ds\int_0^{\infty} \frac{\sin^3(x)}{x^3}
\, dx = \frac{3\pi}{8}$.\\
(Hint: consider $f(z) = e^{3\iota z} - 3e^{\iota z} + 2$. Verify
that for $x\in\R$, $\Im(f(x)) = -4\sin^3(x)$ and that
$\ds\frac{f(z)}{z^3}$ has a pole of order $1$ at $z=0$,
so Lemma 28.4 applies.)\\

\problem{3} Let $a,b\in\R_{>0}$. Prove that
$\ds \int_0^{\infty} \frac{\cos(2ax)-\cos(2bx)}{x^2}\, dx
 = \pi(b-a)$.\\

\noindent {\bf II. Problems from Lecture 30.}
Problems 4--6 below involve the Laplace transform. See Lecture
30, \S 30.5.\\

\problem{4} Let $n\in\Z_{\geq 0}$ and $\ds \varphi(t) = \frac{t^n}{n!}$.
Prove that the Laplace transform $\ds\mathcal{L}\varphi (z) = z^{-n-1}$,
if $\Re(z)>0$.\\

\problem{5} Let $\varphi(t) = e^t$. Prove that $\ds\mathcal{L}\varphi
(z) = \frac{1}{z-1}$, if $\Re(z)>1$.\\

\problem{6} Let $\varphi(t)$ be a continuous function of a real variable
$t\in\R_{\geq 0}$. Let $c\in\R$ and define $\psi(t) = e^{ct}\varphi(t)$.
Prove that $\ds\mathcal{L}\psi (z) = \mathcal{L}\varphi (z-c)$.\\

\noindent {\bf III. Problems from Lectures 31, 32 - Gamma function.}
For the rest of the problems, you can use the following facts.
\begin{enumerate}
\item Definition of $\Gamma(z)$ (Weierstrass). $\Gamma : \C\dashrightarrow
\C$ is a meromorphic function with poles of order $1$ at
$z=0,-1,-2,\ldots$
\[
\Gamma(z) = \frac{1}{ze^{\gamma z}} \prod_{n=1}^{\infty}
\left\{\lp 1+\frac{z}{n}\rp^{-1} e^{\frac{z}{n}}\right\}
\]
where $\gamma$ is the Euler--Mascheroni constant (see Lecture 32,
\S 32.1--32.3).\\

\item For $\Re(z)>0$, $\ds \Gamma(z) = \int_0^{\infty}
t^{z-1}e^{-t}\, dt$ (Euler's integral).\\

\item $\ds \Gamma(z+1) = z\Gamma(z)$. $\Gamma(n)=(n-1)!$, for $n\in\Z_{\geq 1}$.
(These facts were proved in Lecture 31, \S 31.3, 31.4, using Euler's integral.
It will be beneficial to prove these directly from Weierstrass' formula
- see Problem 8 below.)\\

\item $\ds \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$
(see Lecture 32, \S 32.5).\\

\end{enumerate}

\problem{7} Prove that, for every $n\in\Z_{\geq 0}$, we have
$\ds\Res{z=-n}{\Gamma(z)} = \frac{(-1)^n}{n!}$.\\

\problem{8} Consider the function (sometimes called Gauss' Psi-function)
\[
\boxed{\Psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}}
\]

\begin{itemize}
\item[(a)] Verify the following formula, using Weierstrass' formula
for the Gamma function:
\[
\Psi(z) = -\gamma -\frac{1}{z} - \sum_{n=1}^{\infty}
\lp \frac{1}{z+n}-\frac{1}{n}\rp\ .
\]

\item[(b)] Use (a) to prove that $\ds\Psi(z+1)-\Psi(z)=\frac{1}{z}$.
Deduce that $\Gamma(z+1)=z\Gamma(z)$.\\
{\em This proves that Weierstrass' formula also solves $F(z+1)=zF(z)$.
Unlike the proof in the notes, this proof avoids the use of
Euler's integral.}\\

\item[(c)] Prove that $\Psi(z)-\Psi(1-z)=-\pi\cot(\pi z)$.\\
\end{itemize}

\problem{9} Let $n\in\Z_{\geq 1}$. Prove that
$\ds \Gamma'(n) = (n-1)!\lp-\gamma+\sum_{k=1}^{n-1}\frac{1}{k}\rp$.\\
(Hint: use Problem 8 (a).)\\

\problem{10} Use $\Psi(z)$ from Problem 8 to find a function $F(z)$
such that:
\[
F(z+1) = F(z) - \frac{1}{(z-2)^2}\ .
\]

\problem{$10^\ast$} Let $P(z),Q(z)$ be two polynomials. Obtain
a method to solve the equation: $\ds F(z+1)-F(z)=\frac{P(z)}{Q(z)}$.\\

\problem{11} Use $\Gamma(z)$ to solve the following equation:
\[
F(z+1) = \frac{z^2-2z}{(z+\iota)^3} F(z)\ .
\]

\problem{$11^\ast$} Let $P(z),Q(z)$ be two polynomials. Obtain a solution
of 
\[F(z+1)=\frac{P(z)}{Q(z)} F(z).
\]

\noindent {\bf Bonus 1.} Let $y\in\R_{\neq 0}$. Prove that
\[
\left|\Gamma\lp\iota y\rp\right| = \sqrt{\frac{2\pi}{y(e^{\pi y}
-e^{-\pi y})}}
\]
(Hint: use the expansion of $\ds\frac{\sin(z)}{z}$ from Lecture 29,
page 1, and the fact that $\sin(\iota y) = \iota (e^y - e^{-y})/2$.)\\

\noindent {\bf Bonus 2.}\footnote{This fascinating computation is due to
Dirichlet, around 1829.} 
Let $f(t)$ be a continuous function of one real
variable $t$. Let $T$ be the triangular region in $\R^2$, given by:
\[
T = \{(x,y) : x\leq 0, y\leq 0 \text{ and } x+y\leq 1\}
\]
Prove that, for any two $a,b\in\R_{>0}$, we have:
\[
\iint_T f(x+y)x^{a-1}y^{b-1}\ dxdy = 
\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
\int_0^1 f(t) t^{a+b-1}\, dt
\]
{\em Hint:} (1) Write this integral as: 
\[
\int_0^1 \lp \int_0^{1-x} f(x+y)x^{a-1}y^{b-1} dy \rp dx\ .
\]
(2) Perform the change of variables: $y = x(1-v)/v$.\\
(3) Flip the order of integration.\\
(4) Change variables again: $vt = x$.\\
(5) Use the result from Lecture 31, \S 31.6.

\end{document}
