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\title{Mathematics 311:  Complex Analysis --- Assignment 6}
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{\bf Reading}:  Marsden, section 3.3.

{\bf Problems}:
1.  Let $f(z)$ be analytic in the entire complex plane $\C$, 
and set $M(r)=\sup_{|z|=r}|f(z)|$.
Show that $M(r_1)\le M(r_2)$ whenever $r_1<r_2$, 
and determine when equality can occur.

2.  Find all expansions in powers of $z$ of the function
$$
f(z)=\frac1{z^2(z^2+1)(z^2+9)}.
$$
(Note that $z^2$ is already a power of~$z$.  Use partial fractions and
then the handy formulas from class.)

3.  Find all expansions in powers of~$z-1$ and in
powers of~$z+1/2$ of the function
$$
f(z)=\frac1{(2z+1)(z-1)}.
$$
(There is no need to use partial fractions on this problem.)

4.  Suppose $f(z)$ is analytic in the strip $a<y<b$ (where $z=x+iy$) 
and satisfies $f(z+1)=f(z)$.
Show that $f(z)$ has a complex Fourier expansion
$$
f(z)=\sum_{n=-\infty}^\infty a_ne^{2\pi inz}
$$
converging at all points of this strip, where
$$
a_n=e^{\pi n(a+b)}\int_{x=0}^1e^{-2\pi inx}f(x+\frac{a+b}2i)\,\dx.
$$
(Suggestion:  Consider the mapping $w=e^{2\pi iz}$.  Show that $f(z)$ 
determines an analytic function of $w$ in the appropriate region, 
and use the two-sided expansion.  Indicate where the hypothesis
$f(z+1)=f(z)$ is needed.)

5.  Find all singular points (either in the finite complex plane~$\C$
or at~$\infty$) of the following functions, and for each one indicate
whether it is nonisolated, essential, a pole, or removable.

(a) $f(z)=1/[(2z+1)(z-1)]$; 

(b) $f(z)=1/(e^z-1)$; 

(c) $f(z)=\pi/\sin(\pi/z)$.

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