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Math 0250 Homework 1

0.1 Number 4
 a
 b

 c

 d

 e

0.2 Number 1
 a
 The point 34i is in the fourth quadrant, so you can use
a polar angle between and , given by
. It's distance from the origin is
. Therefore
34i = 5 e^{0.92730 i}
 b
 The point 5+12i is in the second quadrant, so you can use
a polar angle between and , given by
. It's distance from the origin is
. Therefore
5+12i = 13 e^{1.9656 i}
 c
 The point 3+4i is in the fourth quadrant, so you can use
a polar angle between and , given by
. It's distance from the origin is
. Therefore
34i = 5 e^{0.92730 i}

0.2 Number 4
 a
 b

 c

0.2 Number 5
 a
 Write
 b
 Write
Taking k = 0, 1, 2 gives the three values

C1
The computer gives for because .

C2
The value of b is b = e^{10i} = 0.83907  0.54402i and its log is
. The reason you didn't get 10i for the log is that
the computer represents e^{10i} with a polar angle between and
before taking the log, and so always returns a log with imaginary
part between and . The number 10i is one of infinitely many
value of . The value the computer gives is .

C3
You should have gotten a bunch of points arranged around a circle of radius
one and centered at the origin. That's because always sits on this circle.

C4
You should have gotten a long list of 1's. That's because you're taking
the absolute values of a bunch of points, each of which is exactly one
unit away from the origin, and absolute value measures the distance from
the origin.

C5
You should have gotten a bunch of points arranged on a vertical line, to
the right of the imaginary axis by units. That's because you're
taking the log of a bunch of numbers of the form 2 e^{it}, which gives
a bunch of numbers of the form , each with real part
.

C6
Here's the ``convincing argument''. Write
with r=A and . Then
Taking real parts gives the formula. Are you convinced?

C7
Just calculate. With A = abi,
which has real part as required.
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Frank Beatrous
1/16/1998