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Math 311 Assignment #6 (Also a sample midterm)
Due Oct. 24, 2011
Some information on the midterm:
• Time and location: 9:00-9:50, Nov. 2, 2011, DP 1030
• Sections covered by the midterm: (BC) Sec. 1-25, 27, 29-36
(BMPS) 1.0-1.5, 2.1-2.4, 3.4-3.5
(1) (20 points) Determine which of the following functions f (z) are
entire and which are not? You must justify your answer. Also
find the complex derivative f 0 (z) of f (z) if f (z) is entire. Here
z = x + yi with x = Re(z) and y = Im(z).
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(a) f (z) =
1 + |z|2
z
(b) f (z) = 2(3 ) (here 2z and 3z are taken to be the principal
values of 2z and 3z , respectively, by convention)
(c) f (z) = (x2 − y 2 ) − 2xyi
(d) f (z) = (x2 − y 2 ) + 2xyi
(2) (20 points) Let CR denote the upper half of the circle |z| = R
for some R > 1. Show that
eiz
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z 2 + z + 1 ≤ (R − 1)2
for all z lying on CR .
(3) (20 points) Find all the complex roots of the equation cos z = 3.
(4) (20 points) Find ii and its principal value.
(5) (20 points) Let
(
z 2 /|z| if z =
6 0
f (z) =
0
if z = 0
Show that f (z) is continuous everywhere but nowhere analytic
on C.
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