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\title{Mathematics 311:  Complex Analysis --- Assignment 3}
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{\bf Reading}:  Marsden, sections 2.1, 2.2, 2.3.

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{\bf Problems}:

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1.  Evaluate $\int_{-i}^{i}|z|\,\dz$ by taking the path (a) rectilinearly, 
(b) along the left half of the unit circle, 
(c) along the right half of the unit circle.

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2.  Compute $\int_{|z|=1}\overline{z}\,\dz$.  
(By convention, when a simple closed path is specified as a point set, 
traverse it once counterclockwise.)

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3.  Compute $\int_\gamma\dz/(z^2-1)$, where $\gamma$ is the closed curve
comprising the right half of $\{|z|=2\}$ and $\{0+iy:-2\le y\le2\}$,
traversed once counterclockwise.  (Hint:  partial fractions and the 
Deformation Theorem.)

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4.  Suppose that $\gamma$ is a simple closed curve, and that $f(z)$ 
is an analytic function in a region containing $\gamma$; 
suppose moreover that the derivative $f'(z)$ is continuous in that
region.  Show that $\int_\gamma\overline{f(z)}f'(z)\,\dz$ is purely imaginary.

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5.  Recall from advanced calculus Green's theorem:  if $\gamma$ is the smooth
oriented boundary of an open subset $\Omega$ of the plane and if $u$, $v$ 
are continuously differentiable real-valued functions in an open 
neighborhood of the closure of $\Omega$ then
$$
\int_\gamma u\,\dx+v\,\dy = \int_\Omega\left(\frac{\partial v}{\partial x} -
\frac{\partial u}{\partial y}\right)\,\dx\,\dy.
$$

(a) Show that Green's theorem can be restated:  if $\gamma$, $\Omega$ 
are as above and $f$ is a complex-valued continuously differentiable 
(with respect to $x$ and $y$) function in an open neighborhood of the 
closure of $\Omega$ then
$$
\int_\gamma f(z)\,\dz 
= 2i\int_\Omega\frac{\partial f}{\partial\overline{z}}\,\dx\,\dy,
$$
where as usual, $\partial/\partial\overline{z}
=1/2(\partial/\partial x +i\,\partial/\partial y)$.

(b) Use Green's theorem as restated in part~(a) to prove Cauchy's
theorem under the hypothesis that $f(z)$ is analytic AND has
continuous partial derivatives with respect to $x$ and~$y$.  Is this
result stronger or weaker than the complex-analytic statement of Cauchy's
theorem?

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