.. 1990 Mar 23
.. ===========

Examination 2
=============

1990 Mar
--------

**Instructions.** There are 5 questions: Each question will be graded on a scale of 0-10. A passing score at the Ph.D. level is 30. A passing score at the M.A. level is 20.

**Notation.** The symbol :math:`D` stands for the unit disc :math:`\{z \in \mathbb C \mid | z | < 1\}`.

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.. index:: Riemann mapping theorem

.. _1990May_1:

.. proof:prob::

   (a) State the Riemann mapping theorem (including the relevant topological definitions).
   
   (b) Present a conformal mapping :math:`f` of :math:`D` onto :math:`\Omega = \{z= x + i y \mid x > 0, y > 0\}` such that :math:`f(0)= 1 + i`.

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

.. index:: residue theorem

.. _1990May_2:

.. proof:prob::

   Evaluate :math:`\int_{-\infty}^\infty \frac{dx}{1+x^2}` by use of the residue theorem.

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

.. _1990May_3:

.. proof:prob::

   Suppose that :math:`f` is holomorphic on :math:`D`, :math:`|f(z)| < 1` for all :math:`z\in D`, and :math:`f(1/2) = 1/2`.
   
   (a) Prove that :math:`|f'(1/2)|\leq 1`.
   
   (b) Determine :math:`f(z)` if :math:`f'(1/2) = 1`.

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

.. _1990May_4:

.. proof:prob::

   Let :math:`\{f_n(z)\}` be a sequence of nonvanishing analytic functions on a region :math:`\Omega`. Suppose that :math:`\{f_n(z)\}` converges uniformly on every compact subset of :math:`\Omega` to a function :math:`f(z)`. If :math:`f(z_0) \neq 0` for some :math:`z_0 \in \Omega`, then prove that :math:`f(z)\neq 0` for all :math:`z\in \Omega`.

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

.. _1990May_5:

.. proof:prob::

   Suppose that :math:`U(z)` is a real valued harmonic function defined for all :math:`z \in \mathbb C`. If

   .. math:: \lim_{y\to 0+}\frac{u(x+iy)}{y} = O

   for all :math:`x \in \mathbb R`, then prove that :math:`u(z) = 0` for all :math:`z \in \mathbb C`.

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Solutions
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Solution to :numref:`Problem {number} <1990May_1>`
(coming soon)

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Solution to :numref:`Problem {number} <1990May_2>`
(coming soon)

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Solution to :numref:`Problem {number} <1990May_3>`
(coming soon)

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Solution to :numref:`Problem {number} <1990May_4>`
(coming soon)

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Solution to :numref:`Problem {number} <1990May_5>`
(coming soon)

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