D
efinition
1.8. A sequence
{
z
n
}
converges to
w
∈
C
if for every
ε >
0 there exists a number
N
=
N
ε
>
0 such that

z
n

w

< ε
as soon as
n
>
N
. Notation: lim
n
→∞
z
n
=
w
or
z
n
→
w
,
n
→ ∞
.
A sequence
{
z
n
}
is said to be a Cauchy sequence (or Cauchy), if for every
ε >
0 there is a
number
N
=
N
ε
>
0 such that

z
n

z
m

< ε
as soon as
n
,
m
>
N
. In other words,

z
n

z
m
 →
0
as
n
,
m
→ ∞
.
6
1. GEOMETRY, TOPOLOGY AND ANALYSIS IN THE COMPLEX PLANE
This definition is exactly the same as in Real Analysis except that the sequence and the
limit are now complexvalued.
P
roposition
1.9.
z
n
→
w i
ff
Re
z
n
→
Re
w and
Im
z
n
→
Im
w as n
→ ∞
.
P
roof
. Write:

Re
z
n

Re
w

2
+

Im
z
n

Im
w

2
=

z
n

w

2
.
The righthand side tends to zero as
n
→ ∞
, and the lefthand side is nonnegative. Therefore,
the lefthand side and the righthand side tend to zero at the same time, as required.
C
orollary
1.10.
If z
n
→
w, then
z
n
→
w and

z
n
 → 
w

.
P
roposition
1.11.
A sequence
{
z
n
}
converges i
ff
it is Cauchy.
P
roof
. The realvalued sequences Re
z
n
, Im
z
n
converge i
ff
they are Cauchy, see Analysis
1.
In view of this proposition we say that the set of complex numbers
C
is complete.
2.2. Sets of the complex plane.
D
efinition
1.12. Let
z
0
be a complex number and
r
>
0 be a real number. Then the set
γ
(
z
0
,
r
)
=
{
z
:

z

z
0

=
r
}
=
{
z
=
z
0
+
re
i
φ
, φ
∈
(

π, π
]
}
,
is called circle centered at
z
0
∈
C
of radius
r
>
0. The set
D
(
z
0
,
r
)
=
{
z
∈
C
 
z

z
0

<
r
}
is called an
r

neighbourhood
of
z
0
or an
open disk
of radius
r
centered at
z
0
. The set
D
(
z
0
,
r
)
=
{
z
∈
C
 
z

z
0
 ≤
r
}
is called the closed disk of radius
r
centered at
z
0
.
The set
D
0
(
z
0
,
r
)
=
{
z
∈
C
: 0
<

z

z
0

<
r
}
is called a punctured
r
neighbourhood.
The set
Π
±
=
{
z
:
±
Im
z
>
0
}
is called the
upper (lower) halfplane
.
Let
S
⊂
C
be a set of complex numbers.
D
efinition
1.13. A point
z
∈
S
is said to be an interior point of the set
S
if there is a
r
>
0
such that
D
(
z
,
r
)
⊂
S
.
The notation int
S
is used for the set of all interior points of
S
.
The set
S
is said to be open if
S
=
int
S
, that is
S
consist of interior points, that is for each
z
∈
S
there is a number
r
>
0(possibly depending on
z
) such that
D
(
z
,
r
)
⊂
S
.
3. FUNCTIONS, THEIR LIMITS AND CONTINUITY
7
E
xample
.
(1) The set
Π
+
is open.
Indeed, let
z
=
x
+
iy
∈
Π
+
,
y
>
0.
For any
w
∈
D
(
z
,
y
) we have
Im
w
=
y
+
Im(
w

x
)
≥
y
 
w

x

>
0
,
and hence
w
∈
Π
+
.
(2) The open disk
D
(
a
,
r
) is open. Indeed, let
z
∈
D
(
a
,
r
). Denote
ε
=
r
 
z

a

>
0.
Then for any
w
∈
D
(
z
, ε
) we have

w

a
 ≤ 
w

z

+

z

a

< ε
+

z

a

=
r
,
as required.
(3) Let
a
,
b
∈
C
. The set
Ω =
{
z
∈
C
: Im
z
≥
0
}
is not open. Take
w
=
1
∈
Ω
. For every
ε >
0 the disk
D
(1
, ε
) contains the point 1

i
ε/
2, which does not belong to
Ω
.
This set is in fact closed.
3. Functions, their limits and continuity
3.1. Functions.
We have already met functions in real variable theory and in set theory.
Recall that every function,
f
, comes with a
domain
D
(
f
) and a
range
R
(
f
). Then
f
maps the
members of the domain to the members of the range. More precisely,
f
maps a single member
of the domain to a single member of the range. Functions
do not
map one member of a domain
to more than one member of the range.
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 Spring '20
 Exponential Function, Taylor Series, Complex number, Complex Plane