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Complex Numbers for Fourier
Complex Numbers for Fourier
5 selected
Difficulty 3-7
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Foundation
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Question 1 of 5
120s
foundation (3/10)
compute
Using Euler's formula
e
i
θ
=
cos
θ
+
i
sin
θ
, which of the following equalities is correct?
Hide and think first
A.
e
i
θ
=
θ
(
cos
1
+
i
sin
1
)
when
θ
is small, by linearizing the exponential around
θ
=
0
.
B.
e
iπ
=
−
1
and
e
iπ
/2
=
i
, with
∣
e
i
θ
∣
=
1
for every real
θ
since
cos
2
θ
+
sin
2
θ
=
1
.
C.
e
iπ
=
0
because the imaginary part vanishes, and
e
iπ
/2
=
−
i
since rotation by
π
/2
from the positive real axis lands on the negative imaginary axis.
D.
e
2
π
i
=
−
1
because
2
π
is a half-turn around the unit circle, and
∣
e
i
θ
∣
depends on
θ
in a way that grows linearly with the argument.
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