18.04 Practice problems for final exam, Spring 2018
On the final exam you will be given a copy of the Laplace table posted with these problems.
Problem 1.
Which of the following are meromporphic in the whole plane.
(a) �5
(b) �5∕2
(c) e1∕�
(d) 1∕ sin(�).
Problem 2. (� − 2)2�3 � ′(�)
(a) Let � (�) = . Compute ∫|�|=3 � �
(� + 5)3(� + 1)3(� − 1)4 � (�)
(b) Find the number of roots of �(�) = 6�4+ �3− 2�2+ � − 1 = 0 in the unit disk.
(c) Suppose � (�) is analytic on and inside the unit circle. Suppose also that |� (�)| < 1 for |�| = 1.
Show that � (�) has exactly one fixed point � (�0) = �0 inside the unit circle.
(d) True or false: Suppose � (�) is analytic on and inside a simple closed curve �. If � has � zeros
inside � then � ′(�) has � − 1 zeros inside �.
Problem 3.
Let � = {�| 0 ≤ Re(�) ≤ �∕2, Im(�) ≥ 0.
Let � = the first quadrant/
Show that � (�) = sin(�) maps � conformally onto �
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