MAT 3730Complex Variables

Section 2.2

Limits and Continuity

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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

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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