Question 1
Let
f
(
z
)
f(z)f, left parenthesis, z, right parenthesis be defined on
C
∖
{
0
}
C∖{0}C, \setminus, left brace, 0, right brace via
f
(
z
)
=
ln
∣
z
∣
+
i
θ
f(z)=ln∣z∣+iθf, left parenthesis, z, right parenthesis, equals, natural log, vertical bar, z, vertical bar, plus, i, theta, where
θ
θtheta is the argument of
z
zz satisfying
0
≤
θ
<
2
π
0≤θ<2π0, is less than or equal to, theta, \lt, 2, pi (note that this is not the principal argument). Which of the following are true: (Check all that apply.)
f
(
i
2
)
−
2
f
(
−
i
)
=
−
2
π
i
f(i
2
)−2f(−i)=−2πif, left parenthesis, i, squared, right parenthesis, minus, 2, f, left parenthesis, minus, i, right parenthesis, equals, minus, 2, pi, i.
The image under
f
ff of the circle of radius
e
ee, centered at the origin, is the entire vertical line
{
x
=
1
}
{x=1}left brace, x, equals, 1, right brace.
f
′
(
z
)
=
1
z
f′(z)=
z
1
f, prime, left parenthesis, z, right parenthesis, equals, start fraction, 1, divided by, z, end fraction for
z
∈
C
∖
[
0
,
∞
)
z∈C∖[0,∞)z, \in, C, \setminus, open bracket, 0, comma, infinity, right parenthesis.
f
ff is continuous at
z
=
2
z=2z, equals, 2.
1 point
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