18.6 Review of the Second Part of the Lecture
18.6.1 Final Exam
The final exam of MATH 529 will take place on
Tuesday, May 3, 2011, 8:00
– 11:00 AM,
in Phillips Hall, Room 332 (the regular lecture hall).
Please
read the paragraph on “Final Examinations” in the Academic Procedures
section of the 2010–2011 Undergraduate Bulletin
(http://www.unc.edu/ugradbulletin/procedures1.html), to be informed about
what to do in case you are unable to take the exam at the scheduled time.
As stated in the course syllabus, you are allowed to use a summary of the
lecture notes on 6 (six) pages (US letter format, doublesided) for the final
exam. This is meant as 6 sheets of paper, with text on both sides.
The exam will cover topics from Chapter 12 (up to and including 12.10)
and 13–17 (not 16.4).
The problems will focus on the
application
of the
methods treated in this course; I want to see that students are able to recog
nize which method should be used for a given problem, and that they apply
it correctly. I will not ask for proofs, and I will try to avoid complicated aux
iliary calculations (such as partial fraction decompositions, finding roots of
high order polynomials, or computing highorder derivatives of complicated
functions). There will be
∼
9 problems in the final exam, which means
∼
20
minutes to solve each problem.
18.6.2 Complex Numbers and Functions (13)
Complex Numbers. Complex Plane
We had constructed the field of
complex numbers
C
by considering ordered pairs (
x, y
) of real numbers. A
complex number
z
∈
C
may be represented in various ways in the complex
plane
C
≃
R
2
, such as
z
=
x
+
iy
=
r
(cos
ϕ
+
i
sin
ϕ
) =
re
iϕ
,
i
2
=
−
1
,
(1067)
where
x
:= Re
z
denotes the
real part
of
z
,
y
:= Im
z
the
imaginary part,
r
=

z

=
√
zz
=
radicalbig
(Re
z
)
2
+ (Im
z
)
2
the
absolute value,
and
ϕ
= arg
z
the
argument
of
z
. It is defined by the equations
cos
ϕ
=
x
r
,
sin
ϕ
=
y
r
,
(1068)
and therefore multivalued, because of the periodicity of sin and cos.
We
typically restrict the argument to either arg
z
∈
[0
,
2
π
) or Arg
z
∈
(
−
π, π
].
198
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The latter is convenient when working with complex logarithms, because
Arg
z
= Im(Log
z
), and the branch cut of Arg coincides with the branch cut
of Log. We have
arg
z
=
braceleftbigg
Arg
z,
Im
z
≥
0
2
π
+ Arg
z,
Im
z <
0
,
z
negationslash
= 0
.
(1069)
The complex conjugate
z
∈
C
of a complex number
z
∈
C
is defined by
z
= Re
z
−
i
Im
z
; geometrically, this corresponds to a mirroring on the real
axis in the complex plane, and therefore
z
=
z
⇔
Im
z
= 0. The absolute
value

z

=
√
zz
induces a metric on
C
,
d
:
C
×
C
→
C
,
d
(
z
1
, z
2
) :=

z
1
−
z
2

.
Thus we can measure distances between complex numbers (Problem Set 7).
In particular, we have the triangle inequality

z
1
+
z
2
 ≤ 
z
1

+

z
2

,
z
1
, z
2
∈
C
.
Polar Form of Complex Numbers. Powers and Roots.
The polar
form of a complex number is useful to represent products, quotients, powers
and roots of complex numbers:
z
1
z
2
=

z
1

z
2

exp (
i
(arg
z
1
+ arg
z
2
))
,
(1070)
z
1
z
2
=

z
1


z
2

exp (
i
(arg
z
1
−
arg
z
2
))
,
(1071)
z
n
=

z

n
exp (
in
arg
z
)
,
(1072)
n
√
z
=
n
radicalbig

z

exp
parenleftBig
i
arg
z
n
parenrightBig
exp
parenleftbigg
i
2
πk
n
parenrightbigg
,
k
= 0
, . . . , n
−
1
.
(1073)
Derivative. Holomorphic Function.
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 Spring '08
 Staff
 Math, Power Series, Complex Numbers, Complex number

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