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\title{Analysis II}
\author{Homework 1}
\date{Due in class Monday, February 11, 2019}
\maketitle

\begin{enumerate}

\item Recall the mean value theorem of integral calculus: Suppose
$g \colon [a,b] \to \R$ is Riemann integrable and non-negative, and
$f \colon [a,b] \to \R$ is continuous.  Then there exists
$\xi \in [a,b]$ such that
\[
  f(\xi) \int_a^b g(x) \, \d x
  = \int_a^b f(x) \, g(x) \, \d x \,.
\]
Give an example each to show that the following assumptions cannot be
generally dropped:
\begin{enumerate}
\item $g$ is non-negative,
\item $f$ is continuous.
\end{enumerate}

\item Recall Taylor's theorem in the following form.  Suppose $f \in
C^{n+1}(I)$ for some open interval $I$.  Then for all $a,x \in I$,
\[
  f(x) = \sum_{j = 0}^n \frac{f^{(j)}}{j!} \, (x-a)^j
         + R_n(x)
\]
where
\[
  R_n(x) = \frac1{n!} \int_a^x (x-t)^n \, f^{(n+1)}(t) \, \d t \,.
\]
\begin{enumerate}
\item Turn the derivation shown in class into a formal proof by
induction.
\item Show that the remainder can also be written as
\[
  R_n(x) = \frac{(x-a)^{n+1}}{n!}
  \int_0^1 (1-s)^n \, f^{(n+1)}(a + s(x-a)) \, \d s \,.
\]
\item Show that there exists $\xi \in [a,x]$ such that
\[
  R_n(x) = \frac{(x-a)^{n+1}}{(n+1)!} \, f^{(n+1)} (\xi) \,.
\]
(This expression is known as the \emph{Lagrange form} of the
remainder.) 
\end{enumerate}

\item Show that the \emph{gamma function}
\[
  \Gamma(x) = \int_0^\infty t^{x-1} \, \e^{-t} \, \d t
\]
is defined for all $x>0$ and satisfies $\Gamma(x+1) = x!$ for each
$x\in \mathbb N$.

\emph{Hint:} Use integration by parts to show that $\Gamma(x+1) = x \,
\Gamma(x)$. 

\item Verify the value of the \emph{Gaussian integral}
\[
  \int_{-\infty}^{\infty} \e^{-x^2} \, \d x = \sqrt \pi \,.
\]

\emph{Hint:} There are many different proofs, most of which use
multi-dimensional integration which we did (yet) cover.  An elegant
derivation which avoids all multi-dimensional integrals works along
the following lines.

\begin{enumerate}
\item Let
\[
  F(t) = \int_0^\infty \frac{\e^{-t^2(1+x^2)}}{1+x^2} \, \d x
\]
and show that $F(0)=\frac\pi2$ and $\lim_{t\to\infty} F(t) = 0$.

\item Let
\[
  J = \int_0^\infty \e^{-x^2} \, \d x
\]
and show that $F'(t) = -2 \, J \, \e^{-t^2}$.  (Fully justify your
manipulations!)

\item Use the Fundamental Theorem of Calculus on the result from part
(b). 

\end{enumerate}
    
\item Check the conditions for Laplace's method for the function
\[
  f(x) = x-\ln x - 1
\]
which appears in the proof of  Stirling's formula.  I.e., check the
following. 
\begin{enumerate}
\item $f$ is strictly decreasing for $0<x<1$, strictly increasing for
$x>1$, and $f(1)=0$.
\item There are positive constants $b$ and $c$ such that $f(x) \geq
b\,x$ for $x \geq c$.
\item $f(x) = a \, (x-1)^2 + \psi(x) \, (x-1)^3$ where $\psi$ is a
bounded function for $x \in [1-\delta,1+\delta]$ for some $\delta >
0$.  Find $a$ explicitly.
\end{enumerate}

\end{enumerate}

\end{document}
