.. 2007 Nov 16
.. ============

Examination 9
=============

2007 Nov
---------

**Instructions.** Do as many problems as you can. Complete solutions (except for minor flaws) to 5 problems would be considered an excellent performance. Fewer
than 5 complete solutions may still be passing, depending on the quality.

.. _2007Nov_1:

.. proof:prob::

   Let :math:`G` be a bounded open subset of the complex plane. Suppose :math:`f` is continuous on the closure of :math:`G` and analytic on :math:`G`. Suppose further that there is a constant :math:`c\geq 0` such that :math:`|f| = c` for all :math:`z` on the boundary of :math:`G`. Show that either :math:`f` is constant on :math:`G` or :math:`f` has a zero in :math:`G`.

.. _2007Nov_2:

.. proof:prob::

   (a) State the residue theorem.

   (b) Use contour integration to evaluate

       .. math:: \int_0^\infty \frac{x^2}{(x^2+1)^2} \, dx.

       *Important:* You must carefully: specify your contours, prove the inequalities that provide your limiting arguments, and show how to evaluate all relevant residues.

.. _2007Nov_3:

.. proof:prob::

   (a) State :ref:`Schwarz's lemma <schwarz>`.

   (b) Suppose :math:`f` is holomorphic in :math:`D = \{z: |z|< 1\}` with :math:`f(D) \subset D`. Let :math:`f_n` denote the composition of :math:`f` with itself :math:`n` times :math:`(n= 2, 3, \dots)`. Show that if :math:`f(0) = 0` and :math:`|f'(0)| < 1`, then :math:`\{f_n\}` converges to 0 locally uniformly on :math:`D`.

.. _2007Nov_4:

.. proof:prob::

   Exhibit a conformal mapping of the region common to the two disks :math:`|z|<1` and :math:`|z-1|<1` onto the region inside the unit circle :math:`|z| = 1`.

.. _2007Nov_5:

.. proof:prob::

   Let :math:`\{f_n\}` be a sequence of functions analytic in the complex plane :math:`\mathbb{C}`, converging uniformly on compact subsets of :math:`\mathbb{C}` to a polynomial :math:`p` of positive degree :math:`m`. Prove that, if :math:`n` is sufficiently large, then :math:`f_n` has at least :math:`m` zeros (counting multiplicities). Do not simply refer to :ref:`Hurwitz's theorem <hurwitz>`; prove this version of it.

.. _2007Nov_6:

.. proof:prob::

   Let :math:`(X, d)` be a metric space.

   (a) Define what it means for a subset :math:`K\subset X` to be compact.

   (b) Prove (using your definition in (a)) that :math:`K\subset X` is compact implies that :math:`K` is both closed and bounded in :math:`X`.

   (c) Give an example that shows the converse of the statement in (b) is false.

-----------------------

Solutions
---------

(coming soon)

.. .. container:: toggle

..   .. container:: header

..      :premium:`Solution 1.`

..   (coming soon)

.. .. container:: toggle

..   .. container:: header

..      :premium:`Solution 2.`

..   (coming soon)

.. .. container:: toggle

..   .. container:: header

..      :premium:`Solution 3.`

..   (coming soon)

.. .. container:: toggle

..   .. container:: header

..      :premium:`Solution 4.`

..   (coming soon)

.. .. container:: toggle

..   .. container:: header

..      :premium:`Solution 5.`

..   (coming soon)

.. .. container:: toggle

..   .. container:: header

..      :premium:`Solution 6.`

..   (coming soon)

.. -------------------------------