mat 3730 complex variables section 2.2 limits and continuity
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MAT 3730Complex Variables
Section 2.2
Limits and Continuity
http://myhome.spu.edu/lauw
Preview
We will follow a similar path of real variables to define the derivative of complex functions
We will look at the conceptual idea of taking limits without going into the precise definition
(MAT 3749 Intro Analysis)
Default
All coefficients are complex numbers All functions are complex-valued
functions
Recall (Real Variables)
Functions Limits DerivativesContinuity
Complex Variables
Functions Limits DerivativesContinuity
Analyticity
Recall: Limits of Real Functions
Left hand Limit
Right Hand Limit
Limit
Left-Hand Limit (Real Variables)
x
y
3
2
The left-hand limit is 2 when x approaches 3
Notation:
Independent of f(3)y=f(x)
2)(lim3
xf
x
Left-Hand Limit (Real Variables)
We write
and say “the left-hand limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close as we like) by taking x to be sufficiently close to a and x less than a.
lim ( )x a
f x L
Right-Hand Limit (Real Variables)
x
y
2
4
The right-hand limit is 4 when x approaches 2
Notation:
Independent of f(2)
y=f(x)
4)(lim2
xfx
Right-Hand Limit (Real Variables)
We write
and say “the right-hand limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close as we like) by taking x to be sufficiently close to a and x greater than a.
lim ( )x a
f x L
Limit of a Function(Real Variables)
if and only if
and
Independent of f(a)
Lxfax
)(lim
Lxfax
)(lim Lxfax
)(lim
Observations
The limit exists if the limits along all possible paths exist and equal
The existence of the limit is independent of the existence and value of f(a)
Lxfax
)(lim
Definition (Conceptual)
w0 is the limit of f(z) as z approaches z0 ,
if f(z) stays arbitrarily close to w0 whenever z is sufficiently near z0
x u
y v
0z 0w
f
zw
Observations (Complex Limits)
The limit exists if the limits along all possible paths exist and equal• There are infinitely many paths
• 2 paths with 2 different limits prove non-existence
• Use the precise definition to prove the existence
The existence of the limit is independent of the existence and value of f(z0)
0)(lim0
wzfzz
Observations (Complex Limits)
The limit exists if the limits along all possible paths exist and equal• There are infinitely many paths
• 2 paths with 2 different limiting values prove non-existence
• Use the precise definition to prove the existence
The existence of the limit is independent of the existence and value of f(z0)
0)(lim0
wzfzz
Observations (Complex Limits)
The limit exists if the limits along all possible paths exist and equal• There are infinitely many paths
• 2 paths with 2 different limiting values prove non-existence
• Use the precise definition to prove the existence
The existence of the limit is independent of the existence and value of f(z0)
0)(lim0
wzfzz
Definition
Let f be a function defined in a nhood of z0. Then f is continuous at z0 if
)()(lim 00
zfzfzz
The Usual….
We are going to state the following standard results without going into the details.
Theorem
0 if )(
)(lim (iii)
)()(lim (ii)
)()(lim (i)
then,)(lim and )(lim If
0
0
0
00
BB
A
zg
zf
ABzgzf
BAzgzf
BzgAzf
zz
zz
zz
zzzz
Theorem
0)( if )(
)( (iii)
)()( (ii)
)()( (i)
at continuous also arefunction following the
then ,at continuous are )( and )( If
0
0
0
zgzg
zf
zgzf
zgzf
z
zzgzf
Common Continuous Functions
20 1 2
20 1 2
20 1 2
(i) The constant functions ( )
(ii) ( ) ( )
(iii) The polynomial functions ( )
(iv) The rational functions
,
are contin
nn
nnm
m
C
f z z C
C
a a z a z a z
a a z a z a z
b b z b z b z
uous at the points where the denominator 0
Example 1
2
3
9Evaluate lim
3z i
z
z z i
MAT 3730Complex Variables
Section 2.3
Analyticity
http://myhome.spu.edu/lauw
Preview
We are going to look at • Definition of Derivatives
• Rules of Differentiation
• Definition of Analytic Functions
Definition
Let f be a function defined on a nhood of z0. Then the derivative of f at z0 is given by
provided this limit exists.
f is said to be differentiable at z0
z
zfzzfzfz
dz
dfz
)()(lim 00
000
Example 2
1
Show that for ,
n n
n N
dz nz
dz
(The Usual…) Rules of Differentiation
2
If and are differentiable at , then
( )
( ) if 0( )
f g z
f g z f z g z
cf z cf z
fg z f z g z f z g z
f z g z f z g zfz g z
g g z
Example 3 (a)
11 1 0
1 21 2 1
is differentiable on the whole plan entiree ( )
( 1) 2
n nn n
n nn n
P z a z a z a z a
P z na z n a z a z a
Example 3 (b)2
0 1 22
0 1 2
( )
is differentiable at the points
where the denominator 0
nnm
m
a a z a z a zR z
b b z b z b z
(The Usual…)The Chain Rule
)())(())((
then,)(at abledifferenti is
and ,at abledifferenti is If
zgzgfzgfdz
d
zgf
zg
A function f is said to be analytic on an open set G if it has a derivative at every point of G
f is analytic at a point z0 means
f is analytic on an nhood of a point z0
Definition
0 0
0
is nowhere differentiable.
Find 2 paths such that the limiting values
( ) ( ) lim
are diffe
re t
:
n
z
KEY
f z z
f z z f z
z
Example 4
Example 4
0 0
0
is nowhere differentiable.
Find 2 paths such that the limiting values
( ) ( ) lim
are diffe
re t
:
n
z
KEY
f z z
f z z f z
z
Next Class
Section 2.4
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