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\title{Mathematics 311:  Complex Analysis --- Assignment 4}
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{\bf Reading}:  Marsden, sections 2.4, 2.5.

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{\bf Problems}:

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1.  Evaluate $\int_\gamma\dfrac{ze^z}{z+2i}\,\dz$ in the following
two cases: (a) $\gamma=\{z\in\C:|z|=1\}$, (b) $\gamma=\{z\in\C:|z|=3\}$.

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2.  Evaluate $\int_{|z|=1}e^zz^{-4}\,\dz$.

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3.  Show that for any complex number~$t$,
$$
\frac1{2\pi i}\int_{|z|=3}\frac{e^{zt}}{z^2+1}\,\dz = \sin t.
$$

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4.  Prove Cauchy's inequality:  If $f$ is analytic in an open neighborhood 
of the closed disk $\{\zeta\in\C:|\zeta-z|\le r\}$ 
and if $f$ satisfies $|f(\zeta)|\le M$ whenever $|\zeta-z|=r$ 
then $|f^{(n)}(z)|/n!\le M/r^n$.

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5.  Show that if $f$ is analytic in the entire plane $\C$, and for
some positive real number~$c$ and some nonnegative integer~$n$ and
some positive real number~$r_o$ we have $|f(z)|\le c|z|^n$ for all~$z$
such that $|z|\ge r_o$, then $f$ must be a polynomial of degree at most~$n$.
(Hint:  Since $f$ is represented everywhere by its power series about~$0$,
it suffices to show that $f^{(n+m)}(0)=0$, for all positive integers~$m$,
i.e., that $|f^{(n+m)}(0)|$ is arbitrarily small for any such~$m$.)

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6.  Show that there can not exist any function~$f$ that is analytic in an 
open neighborhood of a point~$z$ and satisfies $|f^{(n)}(z)|/n!>n^n$ for all 
positive integers~$n$.

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