18.04 Practice problems exam 2, Spring 2018
Problem 1. Harmonic functions
(a) Show �(�, �) = �3− 3��2+ 3�2− 3�2 is harmonic and find a harmonic conjugate.
(b) Find all harmonic functions � on the unit disk such that �(1∕2) = 2 and �(�) ≥ 2 for all � in the
disk.
(c) The temperature of the boundary of the unit disk is maintained at � = 1 in the first quadrant,
� = 2 in the second quadrant, � = 3 in the third quadrant and � = 4 in the fourth quadrant. What
is the temperature at the center of the disk
(d) Show that if � and � are conjugate harmonic functions then �� is harmonic.
(e) Show that if � is harmonic then �� is harmonic.
(f) Show that if � is harmonic and �2 is harmonic the � is constant.
(We always assume harmonic functions are real valued.)
Problem 2.
Let � (�) = 1 . Find Laurent series for � on each of the 3 annular regions centered at
(� − 1)(� − 3)
� = 0 where � is analytic.
Problem 3.
Find the first few terms of the Laurent series around 0 for the following.
(a) � (�) = �2 cos(1∕3�) for 0 < |�|.
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(b) � (�) = for 0 < |�| < �. What is �?
e� − 1
Problem 4. ∞
∑ ��
What is the annulus of convergence for 2|�| .
�=−∞
Problem 5.
Find and classify the isolated singularities of each of the following. Compute the residue at each
such singularity.
�3+ 1
(a) �1(�) = �2(� + 1)
(b) �2(�) = 1
e� − 1
(c) �3(�) = cos(1 − 1∕�)
Problem 6.
(a) Find a function � that has a pole of order 2 at � = 1 + � and essential singularies at � = 0 and
� = 1.
(b) Find a function � that has a removable singularity at � = 0, a pole of order 6 at � = 1 and an
essential singularity at � = �.
Problem 7.
True or false. If true give an argument. If false give a counterexample
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