Part D.
COMPLEX ANALYSIS
Major Changes
In the previous edition, conformal mapping was distributed over several sections in the
first chapter on complex analysis. It has now been given greater emphasis by consolidation
of that material in a separate chapter (Chap. 17), which can be used independently of a
CAS (just as any other chapter) or in part supported by the graphic capabilities of a CAS.
Thus in this respect one has complete freedom.
Recent teaching experience has shown that the present arrangement seems to be
preferable over that of the 8th edition.
CHAPTER 13
Complex Numbers and Functions
SECTION 13.1. Complex Numbers. Complex Plane, page 602
Purpose.
To discuss the algebraic operations for complex numbers and the representation
of complex numbers as points in the plane.
Main Content, Important Concepts
Complex number, real part, imaginary part, imaginary unit
The four algebraic operations in complex
Complex plane, real axis, imaginary axis
Complex conjugates
Two Suggestions on Content
1.
Of course, at the expense of a small conceptual concession, one can also start
immediately from (4), (5),
z
x
iy
,
i
2
1
and go on from there.
2.
If students have some knowledge of complex numbers, the practical division rule
(7) and perhaps (8) and (9) may be the only items to be recalled in this section. (But I
personally would do more in any case.)
SOLUTIONS TO PROBLEM SET 13.1, page 606
2.
Note that
z
2
2
i
and
iz
2
2
i
lie on the bisecting lines of the first and
second quadrants.
4.
z
1
z
2
0 if and only if
Re (
z
1
z
2
)
x
2
x
1
y
2
y
1
0
and
Im (
z
1
z
2
)
y
2
x
1
x
2
y
1
0.
Let
z
2
0, so that
x
2
2
y
2
2
0. Now
x
2
2
y
2
2
is the coefficient determinant of
our homogeneous system of equations in the “unknowns”
x
1
and
y
1
, so that this system
has only the trivial solution; hence
z
1
0.
8.
23
2
i
10.
9, 16
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Page 244
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12.
z
1
/
z
2
7/41
(22/41)
i
,
z
1
/
z
2
(
z
1
/
z
2
)
7/41
(22/41)
i
14.
5/13
(12/13)
i
,
5/13
(12/13)
i
16.
3
x
2
y
y
3
,
y
3
18.
Im
[
(1
i
)
8
z
2
]
Im
[
(2
i
)
4
z
2
]
Im
[
2
4
z
2
]
32
xy
SECTION 13.2. Polar Form of Complex Numbers. Powers and Roots,
page 607
Purpose.
To give the student a firm grasp of the polar form, including the principal value
Arg
z
, and its application in multiplication and division.
Main Content, Important Concepts
Absolute value
z
, argument , principal value Arg
Triangle inequality (6)
Multiplication and division in polar form
n
th root,
n
th roots of unity (16)
SOLUTIONS TO PROBLEM SET 13.2, page 611
2.
2(cos
1
_
2
i
sin
1
_
2
), 2(cos (
1
_
2
)
i
sin (
1
_
2
))
4.
1
_
4
_
1
16
2
(cos arctan
1
_
2
i
sin arctan
1
_
2
)
6.
Simplification shows that the quotient equals
3.
Answer:
3(cos
i
sin
).
8.
Division shows that the given quotient equals
_
22
41
_
7
41
i
.
Hence the polar form is
_
1
41
22
2
7
2
(cos arctan
_
7
22
i
sin arctan
_
7
22
).
10.
3.09163,
3.09163. Of course, the problem should be a reminder that the principal
value of the argument is discontinuous along the negative real axis, where it jumps
by 2
.
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