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1. (10pts) Show that u(z,y) = e-3r cos 3y is harmonic and find a
harmonic conjugate.
2. (10pts) Find a biholomorphic map between the unit disk A and the
region
=
3. (20pts) Let a € C be a constant.
(a) Find the inverse Laplace transform of
F(8)=
3+28
using the Bromwich integral.
(b) Find the inverse Laplace transform of
1
s+a
using the Bromwich integral.
(c) Find the inverse Laplace transform of
1
F(s)
(s+a)²
using the Bromwich integral.
(d) Decompose the rational function
1
s+2s²+s
into principal parts and use this together with (b) and (c) to find the
inverse Laplace transform.
4. (20pts) Show that
do
dx =
where O and
1
if>>0
sgn =
-1 if < < 0.
(Hint: First do the case a > 0 and 8>0.)
5. (20pts) Compute
.° where a € (-1,1).
1
6. (20pts) Show that the equation Z4 + z+1 = 0 has one solution in
each quadrant. Prove that all solutions lie inside the open disk
[zec1121<21
2

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