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\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 6}
\date{Due via Gradescope, Wednesday, April 1 before 20:00}
\maketitle

\emph{Note:} Assignments marked (*) will not be graded.  Do not turn
them in.  However, they will be discussed in the tutorial and example
solutions can be found in the appendix of Kantorovitz' book.

\begin{enumerate}

\item *(Kantorovitz, p.~175, Exercise 1.) Let $\displaystyle F(y) =
\int_0^1 \e^{x^2 y} \, \d x$.
\begin{enumerate}
\item Find $F'(0)$.
\item For $y \neq 0$, show that $F$ satisfies the differential
equation
\[
  2 y \, F'(y) + F(y) = \e^y \,.
\]
\end{enumerate}

\item (Kantorovitz, p.~175, Exercise 1.) Let $b>0$. For $f \in
C([0,b])$, define $F_0 (x) = f(x)$ and
\[
  F_n (x) =  \frac{1}{(n-1)!} \int_0^x (x-y)^{n-1} \, f(y) \, \d y
\]
for $x\in[0,b]$ and $n = 1, 2, \dots$.  Show that $F_n \in C^n([0,b])$
with
\[
  F_n^{(k)} = F_{n-k} \quad \text{for } k = 1, \dots, n \,.
\]

\emph{Remark:} This relation shows that if $J$ denotes the
\emph{integration operator} on $C([0,b])$ defined by
\[
  (Jf)(x) = \int_0^x f(y) \, \d y \,,
\]
then
\[
  F_n = J^n f \,.
\]

\item (Kantorovitz, p.~177, Exercise 5)  Let $0<a<b$ and
\[
  F(y) = \int_{a+y}^{b+y} \frac{\e^{xy}}x \, \d x \,.
\]
Calculate $F'(y)$ for $y>0$.

\item *(Adapted from Kantorovitz, Example 4.1.12) Consider the function
\[
  f(x,y) = \frac{x^2-y^2}{(x^2+y^2)^2}
\]
on the rectangle $I = [0,1] \times [\alpha, 1]$ for $\alpha \in
(0,1)$.

\begin{enumerate}
\item Show that
\[
  \int_0^1 f(x,y) \, \d x = - \frac1{1+y^2}
\]
for every fixed $y \in [\alpha,1]$.

\emph{Hint:} Note that
\[
  \frac{x^2-y^2}{(x^2+y^2)^2}
  = \frac{1}{x^2+y^2} + y \, \frac{\partial}{\partial y}
    \frac{1}{x^2+y^2} \,.
\]

\item Note that the result from (a) continuously extends to the unit
square $I = [0,1]^2$ and conclude that
\[
  \int_0^1 \int_0^1 f(x,y) \, \d x \, \d y = - \frac\pi4
\]
while 
\[
  \int_0^1 \int_0^1 f(x,y) \, \d y \, \d x = \frac\pi4 \,.
\]

\item Does this contradict the theorem on the exchange of partial
integration proved in class?  Explain!

\end{enumerate}



\item (Kantorovitz, p.~177, Exercise 7) For $n \in \mathbb N$,
calculate the iterated integral
\[
  \int_0^\pi \int_0^1 x^{2n-1} \, \cos(x^n y) \, \d x \, \d y \,.
\]

\item Show that a set $B \subset \R^n$ has zero content if it has no
more than finitely many limit points.

\item Let $B \subset \R^n$ be a bounded set and define the
\emph{characteristic function} of $B$ by
\[
  \chi_B (x) =
  \begin{cases}
    1 & \text{if } x \in B \,, \\
    0 & \text{if } x \notin B \,.
  \end{cases}
\]
Show that $\chi_B$ is Riemann-integrable if and only if $B$ has
content.
\newpage
\item *(Kantorovitz, p.~177, Exercise 8)
\begin{enumerate}
\item Calculate the iterated integral
\[
  \int_0^1 \int_0^1 \frac{x}{(1+x^2)(1+xy)} \, \d x \, \d y
\]
in two different ways, and prove thereby that
\[
  \int_0^1 \frac{\ln (1+x)}{1+x^2} \, \d x
  = \frac{\pi \ln 2}{8} \,.
\]
\item Conclude that
\[
  \int_0^1 \frac{\arctan x}{1+x} \, \d x
  = \frac{\pi \ln 2}{8} \,.
\]
\end{enumerate}
\end{enumerate}
\end{document}
