Complex Functions Limit and Continuity

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Complex Functions Limit and Continuity. Mohammed Nasser Acknowledgement: Steve Cunningham. Relation between MM (ML) and Vector space. Mathematical Concepts. Mathematical Concepts. Covariance. Variance. Z. Basis. F. 0.

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Complex FunctionsLimit and Continuity

Mohammed Nasser

Acknowledgement:

Steve Cunningham

Mathematical Concepts

Covariance

Variance

Basis

F

0

Set of f1, f2, …, fn is linearly independent if α1 f1+ α2 f2 + …+ αn fn = 0 ,Holds only if each αi=0.

Finite dimensional if there exist maximum n linearly independent elements ; Otherwise it is infinite-dimensional

Basis: can express every f in F in the form

f = α1 f1+ α2 f2 + …+ αn fn

Linear manifold: αf1+βf2 in F

Mathematical Concepts

Covariance

Variance

Length of a vector

Right-angle triangle

Pythagoras’ theorem

|| x || = (x12+ x22 + x32 )1/2

Inner product of a vector with itself = (vector length)2

xTx =x12+ x22 +x32 = (|| x ||)2

x2

||x||

x1

Some Vector Concepts
• Dot product = scalar

x2

|| x || = (x12+ x22)1/2

x1

||x||

||y||

b

y2

q

y1

x

=/2

Orthogonal vectors: xT y = 0

y

Some Vector Concepts
• Angle between two vectors

x

Dot Product inRn

• XTy=yTx

(aX+bZ)Ty=aXTY+bZTY

• XTX=0 ↔ X=0
Real Inner product space

Inner product space: A vector space X over the reals R is an inner product space if and only if there exists a real-valued symmetric bilinear (linear in each argument)

map (.,.), that satisfies:

< , >: X X→R

<x,y>=<y,x>

<x,x>.≥0, and <x, x>=0↔ x=0

<x+z,y>=<x,y>+<z,y>

<kx,y>=k<x,y>

Show that R is itself an inner product space with inner product <x1,x2>=x1 x2

Complex Inner product space

Inner product space: A vector space X over the

reals C is an inner product space if and only if there

exists a real-valued symmetric bilinear (linear in

each argument) map (.,.), that satisfies:

< , >: X X→C

<x,y>=<y,x>

<x,x>.≥0, and <x, x>=0↔ x=0

<x+z,y>=<x,y>+<z,y>

<kx,y>=k<x,y>

Show that C/Z is itself an inner product space with inner product <z1,z2>=z1 z2

Inner product Space

A vector space V on which an inner product is defined is called an inner product space.Any function on a vector space that satisfies the axioms of an inner product defines an inner product on the space. .

There can be many inner products on a given vector space

Example 2

Let u = (x1, x2), v = (y1, y2), and w = (z1, z2) be arbitrary vectors in R2. Prove that<u, v>, defined as follows, is an inner product on R2.

<u, v>= x1y1 + 4x2y2

Determine the inner product of the vectors (-2, 5), (3, 1) under this inner product.

Solution

Axiom 1:<u, v>= x1y1 + 4x2y2 = y1x1 + 4y2x2 =<v, u>

Axiom 2:<u + v, w>=< (x1, x2) + (y1, y2) , (z1, z2) >

=< (x1 + y1, x2 + y2), (z1, z2) .

= (x1 + y1) z1 + 4(x2 + y2)z2

= x1z1 + 4x2z2 + y1 z1 + 4 y2z2

=<(x1, x2), (z1, z2)>+<(y1, y2), (z1, z2) >

=<u, w>+<v, w>

Axiom 4: <u, u>= <(x1, x2), (x1, x2)>=

Further, if and only if x1 = 0 and x2 = 0. That is u = 0. Thus<u, u> 0, and<u, u>= 0 if and only if u = 0.

The four inner product axioms are satisfied,

<u, v>= x1y1 + 4x2y2 is an inner product on R2.

Axiom 3:<cu, v>= <c(x1, x2), (y1, y2)> =< (cx1, cx2), (y1, y2) >

= cx1y1 + 4cx2y2 = c(x1y1 + 4x2y2)

= c<u, v>

The inner product of the vectors (-2, 5), (3, 1) is

<(-2, 5), (3, 1)>= (-2  3) + 4(5  1) = 14

Example

Consider the vector space M22 of 2  2 matrices. Let u and v defined as follows be arbitrary 2  2 matrices.

Prove that the following function is an inner product on M22.

<u, v>= ae + bf + cg + dh

Determine the inner product of the matrices .

Solution

Axiom 1:,<u, v>= ae + bf + cg + dh = ea + fb + gc + hd =<v, u>

Axiom 3: Let k be a scalar. Then

<ku, v>= kae + kbf + kcg + kdh = k(ae + bf + cg + dh) = k<u, v>

Cauchy–Schwarz Inequality

In an inner product space, |<x,y>|2≤ <x,x><y,y>

and the equality sign holds in a strict inner product space if and only if x and y are rescalings of the same vector.

Defining |x||=<x.x>1/2 we can make every inner product a normed space .

Using CSI we can introduce concept of angle, orthogonality, correlation etc into any innerproduct space

Projection theorem holds in this space

Angle between two vectors

In R2 we first define cosθ, then prove C-S inequality

In Rn we first prove C-S inequality

, then define cosθ

Definition

Let V be an inner product space. The angle between two nonzero vectors u and v in V is given by

Angle between two vectors

The dot product in Rn was used to define angle between vectors. The angle  between vectors u and v in Rn is defined by

Normed spaces

Define the notion of the size of f, an

element in F , a vector spacea vector space

Norm || f ||, || ||: F→[0,∞)

1)||f||=0↔f=0

2) ||kf||=|k|||f||

3) ||f|+||g|| <=||f||+||g||

Both R and Z are normed spaces are spaces with | |

Both Rn and are normed spaces are spaces with Euclidean norm,|| ||

The norm of a vector in Rn can be expressed in terms of the dot product as follows

Definition

Let V be an inner product space. The norm of a vector v is denoted ||v|| and it defined by

Norm of a Vector

Generalize this definition:

The norms in general vector space do not necessary have geometric interpretations, but are often important in numerical work.

Example

Consider the vector space M22 of 2  2 matrices. Let u and v defined as follows be arbitrary 2  2 matrices.

It is known that the function <u, v>= ae + bf + cg + dh is an inner product on M22 by Example 2.

The norm of the matrix is

Definition

Let V be an inner product space with vector norm defined by

The distance between two vectors (points) u and v is defined d(u,v) and is defined by

Distance

As for norm, the concept of distance will not have direct geometrical interpretation. It is however, useful in numerical mathematics to be able to discuss how far apart various functions are.

Show that 1. || ||: F→[0,∞) is a continuous function.

2. d(f,g)=||f-g|| is a metric on F

Metric spaces

Put some structure on our spaceF defining nonnegative function d:F χF→R

f2

F

f1

f3

1.d(f1,, f2)=d(f2,, f1), 2) d(f1,, f2)=0 if and only if f1=f2

3, d(f1,, f2)≤ d(f1,, f3) + d(f3,, f2)

Why are metric spaces important?

Allow us to define the distance between functions

Can be able to treat convergence in the space,and limit and continuity of metric space valued functions on metric space.

Completeness – no holes in the space

We want to look at spaces that are very similar to Euclidean space

Can

We

Define

Rate

Of

Change??

Can we talk about best approximations?

Yes if

Mathematical Concepts

Covariance

Variance

Sequence
• Definition. A sequence of complex numbers, denoted , is a function f, such that f: N  C, i.e, it is a function whose domain is the set of natural numbers between 1 and k, and whose range is a subset of the complex numbers. If k = , then the sequence is called infinite and is denoted by , or more often, zn . (The notation f(n) is equivalent.)
• Having defined sequences and a means for measuring the distance between points, we proceed to define the limit of a sequence.

Meaning of Zn Z0

Zn Z0

|zn-z0|=rn 0

Where rn=

, xn x0,,, yn y0

I proved it in previous classes

Geometric Meaning of Zn Z0

zn tends to z0 in any linear or curvilinear way.

Limit of a Sequence
• Definition. A sequence of complex numbers is said to have the limit z0 , or to converge to z0, if for any  > 0, there exists an integer N such that |zn – z0| <  for all n > N. We denote this by
• Geometrically, this amounts to the fact that z0is the only point of znsuch that any neighborhood about it, no matter how small, contains an infinite number of points zn .

Geometric Meaning of Zn Z0

- -

zN+2

z0

zN+1

zn tends to z0 in any linear or curvilinear way.

Example: Convergent Sequence
• Given , choose N=1/ , p=0

1

0

Establish convergence by applying definition

Necessitates knowledge of p.

Cauchy Sequence
• A sequence in a metric space X such that for every , there is an integer N such that if
• A sequence in a complex field C such that for every , there is an integer N such that if
Example Cauchy Sequence
• Given , choose N=2/, p=0

1

0

Establish convergence by applying definition

No need to know p

Theorems and Exercises

Theorem. Show that zn=xn+iyn z0=x0+iy0 if and only if xn x0, yn y0 .

Ex. Plot the first ten elements of the following sequences and find their limits if they exist:

• 1/n +i 1/n
• 1/n2 +i 1/n2
• n +i 1/n
• (1-1/n )n +i (1+1/n)n

Topology

Topology studies the invariant properties of

object under continuous deformations

e

d

8e > 0, 9d > 0 : |y-x| < d) |f(y)-f(x)| < e

Topology

Topology studies the invariant properties of

object under continuous deformations

S

f-1(S)

8 S open ) f-1(S) open

Limit of a Function
• We say that the complex number w0is the limit of the function f(z) as z approaches z0 if f(z) stays close to w0whenever z is sufficiently near z0 . Formally, we state:
• Definition. Limit of a Complex Sequence. Let f(z) be a function defined in some neighborhood of z0 except with the possible exception of the point z0is the number w0if for any real number  > 0 there exists a positive real number  > 0 such that |f(z) – w0|<  whenever 0<|z - z0|< .
Limits: Interpretation

We can interpret this to mean that if we observe points w within a radius  of w0, we can find a corresponding disk about z0such that all the points in the disk about z0are mapped into it. That is, any neighborhood of w0 contains all the values assumed by f in some full neighborhood of z0, except possibly f(z0).

v

y

w = f(z)

w0

z0

u

x

w-plane

z-plane

Complex Functions : Limit and Continuity

f: Ω1

Ω2

Ω1 and Ω2 are domain and codomain respectively.

Let z0 be a limit point of Ω1 , w0 belongs to Ω2 .

Let us take any B any nbd of w0 in Ω2 and take inverse of B, f-1{B}. f-1{B} contains a nbd of z0 in Ω1..

In the case of Continuity the only difference is w0 =f(z0)),

Properties of Limits
• If as z z0, lim f(z)  A, then A is unique

If as z z0, lim f(z)  A and lim g(z)  B, then

• lim [ f(z)  g(z) ] = A  B
• lim f(z)g(z) = AB, and
• lim f(z)/g(z) = A/B. if B  0.
Continuity
• Definition. Let f(z) be a function such that f: C C. We call f(z) continuous at z0 iff:
• F is defined in a neighborhood of z0,
• The limit exists, and
• A function f is said to be continuous on a set S if it is continuous at each point of S. If a function is not continuous at a point, then it is said to be singular at the point.
Test for Continuity of Functions

it is true in a general metric space but not in general topological space.

f: <S1,d1> <S2,d2> is continuous at s in S1.

For all sn s f(sn) f(s)

Note on Continuity
• One can show that f(z) approaches a limit precisely when its real and imaginary parts approach limits, and the continuity of f(z) is equivalent to the continuity of its real and imaginary parts.
Properties of Continuous Functions
• If f(z) and g(z) are continuous at z0, then so are f(z)  g(z) and f(z)g(z). The quotient f(z)/g(z) is also continuous at z0provided that g(z0)  0.
• Also, continuous functions map compact sets into compact sets.
Exercises
• Find domain and range of the following functions and check their continuity:
• f1(z)=z
• f2(z)=|z|
• f3(z)=z2
• f4(z)=
• f5(z)=1/(z-2)
• f6(z)=ez/log(z)/z1./2/cos(z)
Test for Continuity of Functions

it is true in a general metric space but not in general topological space.

f: <S1,d1> <S2,d2> is continuous at s in S1.

For all sn s f(sn) f(s)

Linear Map and Matrices

This isomorphism is basisdependent

Vn is a finite-dimensional vector space. Let v belongs to Vn

v=x1v1+x2v2+--------+xnvn

(x1,

X2,

xn)

---------

Every Vn is isomorphic to Rn

Its Significance??

Linear Map and Matrices

L

This isomorphism is basisdependent

Vn

Wm

=

L(k1v1+k2v2)

k1L(v1)+k2L(v2))

k1v1+k2v2

Vnm, the set of all such L’s is a vector space of dimension nm

Every Vnm is isomorphic to

Its Significance??

Example: Convergent Sequence
• Given , choose N=1/ , p=0

1

0

Establish convergence by applying definition

Necessitates knowledge of p.

Cauchy Sequence
• A sequence in a metric space X such that for every , there is an integer N such that if
Example: Cauchy Sequence
• Given , choose N=2/, p=0

1

0

Establish convergence by applying definition

No need to know p

Cauchy Sequences and Cauchy Sequences
• (Theorem 3.11 in Rudin)
• (a) In any metric space X, every convergent sequence is a Cauchy sequence.
• (b) If X is a compact metric space and if is a Cauchy sequence in X, then converges to some point of X.
• (c) In , every Cauchy sequence converges.
Complete metric spaces
• A metric space in which every Cauchy sequence converges.
• Examples of complete metric spaces:
• All compact metric spaces
• All Euclidean spaces
• All closed subsets of complete metric spaces.