\documentclass[10pt]{amsart}

\input macros

\begin{document}

\title{Mathematics 311:  Complex Analysis --- Assignment 7}
\maketitle

{\bf Reading}:  Marsden, 4.1, 4.2, 4.3, 6.2, 6.3.

{\bf Problems}:

\medskip

1.  Evaluate the following integrals by using residue techniques (in
each case you have a template available from the lecture):

(a) $\int_{-\infty}^{\infty}\dfrac{x^2\,\dx}{x^4+5x^2+6}$, 

(b) $\int_0^\infty\dfrac{x^2\,\dx}{(x^2+a^2)^3}$ where $0<a\in\R$,

(c) $\int_0^\infty\dfrac{\cos ax\,\dx}{1+x^2}$ where $0<a\in\R$, 

(d) $\int_0^{2\pi}\dfrac{\dtheta}{a+b\sin\theta}$ where $a,b\in\R$ and
$a>|b|>0$,

(e) $\int_0^\infty\dfrac{\root3\of x\,\dx}{1+x^2}$.

\medskip

2.  Let $f(z)$ be analytic inside and on an ellipse $\gamma$ with foci 
at $\{-1, 1\}$.
Show that
$$
\frac{z+1}{z-1}\in[-\infty,0]
\iff
z\in[-1,1].
$$
Consequently, one can define $\log\left(\dfrac{z+1}{z-1}\right)$ as a 
single-valued function on~$\gamma$, taking
$$
{\rm arg}\left(\frac{z+1}{z-1}\right)
={\rm arg}(z+1)-{\rm arg}(z-1),
$$
even though neither argument is well defined by itself.  Show that
$$
\int_\gamma f(z)\log\left(\frac{z+1}{z-1}\right)\,\dz 
= 2\pi i\int_{x=-1}^1f(x)\,\dx
$$
where the right hand expression is a real integral, 
i.e., $x$ runs from $-1$ to $1$ in $\R$.  (Begin by deforming the path
of integration to a barbell.  What is the argument of $(z+1)/(z-1)$ on
the top of the bar, and on the bottom?  How does the integrand grow as
the two end-circles shrink, and how does that compare to the length of
the end-circles?  The Residue Theorem plays no role in this problem.)

\medskip

3.  How many roots of the equation $z^4-6z+3=0$ have their modulus 
between~$1$ and~$2$?

\medskip

4.  If $a\in\R$ and $a>e$, show that the equation $e^z=az^n$ has $n$ 
solutions in the disk $\{z:|z|<1\}$.

\end{document}
