Complex FunctionsLimit and Continuity

Mohammed Nasser Acknowledgement:

Steve Cunningham

Statistical Concepts/Techniques

Concepts in Vector space

Variance Length of a vector, Qd. forms

Covariance Dot product of two vectors

Correlation Angle bt.two vectors

Regression and Classification

Mapping bt two vector sp.

PCA/LDA/CCA Orthogonal/oblique projection on lower dim.

Relation between MM (ML) and Vector space

Math

ematical C

on

cepts

Mathematical Concepts

Covariance

Variance

Z

Basis

3f

2f

1f

0

f

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-dimensionalBasis: can express every f in in the form

f = α1 f1+ α2 f2 + …+ αn fn

Linear manifold: αf1+βf2 in

Basis ( Continued)

Mathematical Concepts

Covariance

Variance

Some Vector Concepts

• Dot product = scalar

ii

iT yxyxyxyx

y

y

y

xxx

3

1332211

3

2

1

321yx

|| x || = (x12+ x2

2 + x32 )1/2

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

xT x =x12+ x2

2 +x32 = (|| x

||)2

x1

x2 ||x||

Right-angle triangle

Pythagoras’ theorem

• Length of a vector

1

21 2

1

...n

Tn i i

i

n

y

yx x x xy

y

x y

|| x || = (x1

2+ x22)1/2

x1

x2

2 1

2 1

1 1 2 2

sin cos

sin cos

cos cos( ) cos cos sin sin

x ycos

x y cos

T

T

y yy y

x xx x

yx yx

x y

x y

x y

• Angle between two vectors

Orthogonal vectors: xT y = 0

x

y

=/2

||x||||y||

y2

y1

x

Some Vector Concepts

1

21 2

1

...n

Tn i i

i

n

y

yx x x xy

y

x y

Dot Product in Rn

; n nf R R R

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

1) <x,y>=<y,x>

2) <x,x>.≥0, and <x, x>=0↔ x=0

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

4) <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

1) <x,y>=<y,x>

2) <x,x>.≥0, and <x, x>=0↔ x=0

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

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

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

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

Inner product Space

Example 2Let 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 3:<cu, v>= <c(x1, x2), (y1, y2)> =< (cx1, cx2), (y1, y2) >

= cx1y1 + 4cx2y2 = c(x1y1 + 4x2y2) = c<u, v>

Axiom 4: <u, u>= <(x1, x2), (x1, x2)>= 04 22

21 xx

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.

04 22

21 xx

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 .

hg

fe

dc

bavu ,

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>

4)01()90()23()52( ,09

25

10

32

09

25 and

10

32

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 Rn we first prove C-S inequality

, then define cosθ

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

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

vu

vu,cos

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

(??)cosvu

vu

Angle between two vectors

Normed spacesDefine the notion of the size of f, an

element in , a vector space• Norm || f ||, || ||: →[0,∞)

1)||f||=0↔f=0

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

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

0,f 0 0f f

1 2 1 2f f f f 1 2 1 2,d f f f f

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

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

n mR

Norm of a Vector

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

vv,v

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

) , , ,() , , ,()() , , ,(

2121

22121

nn

nn

xxxxxxxxxxx

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

hg

fe

dc

bavu ,

2 2 2 2, a b c d || u || u u

Different Norms on C/Z

We have already seen mod, | | is norm on C.

Let | |max and | |tax be two functions defined as follow:

| z|max =max{|x|, |y|}, | z|tax =|x| +|y|,

Show that both are norms on C.

Show that | z|max =< |z|=<| z|tax =<2 | z|max

In fact, it can be shown that for a finite-dimensional space norms are equivalent

All norms are not generated from innerproduct.

Distance

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

vv,v

) ,( ),( vuvuvuvu d

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. || ||: →[0,∞) is a continuous function.

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

Metric spaces

• Put some structure on our space defining nonnegative function d:χ→

2f

1f 1 2( , )d f f

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)

f1

f2

f3

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 talk about best approximations?

Can

We

Define

Rate

Of

Change??

Yes if

Can we get the best from data?

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.

knz 1

1nz

Meaning of Zn Z0

|zn-z0|=rn 0

Where rn=2 2

0 0( ) ( )

n nx x y y

, xn x0,,, yn y0

Zn Z0

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 z0 is the only point of zn such that any neighborhood about it, no matter how small, contains an infinite number of points zn .

1nz

.

lim

0

0

naszz

orzz

n

nn

Geometric Meaning of Zn Z0

zn tends to z0 in any linear or curvilinear way.

zN+1

zN+2

- -

z0

Example: Convergent Sequence

• Given , choose N=1/ , p=0

1/ or ( , ) 1/ or 1/n n np n d s s n z z n

10

Establish convergence by applying definitionNecessitates knowledge of p.

Cauchy Sequence

A sequence in a metric space X such that for every , there is an integer N such that if

{ }n

z

, ,m n N ,X n md s s 0

ns

A sequence in a complex field C such that for every , there is an integer N such that if0

, ,m n N n mz z

Example Cauchy Sequence

• Given , choose N=2/, p=0

1 /nnp n

10

Establish convergence by applying definitionNo 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:

i) 1/n +i 1/n

ii) 1/n2 +i 1/n2

iii) n +i 1/n

iv) (1-1/n )n +i (1+1/n)n

Topology

Topology studies the invariant properties ofobject under continuous deformations

For all > 0, there exists > 0 : |y-x| < ) |f(y)-f(x)| <

Topology

Topology studies the invariant properties ofobject under continuous deformations

For all S open, f-1(S) is open

S

f-1(S)

Limit of a Function

• We say that the complex number w0 is the limit of the function f(z) as z approaches z0 if f(z) stays close to w0 whenever 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 z0 is the number w0 if 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 of ww00, we can find a , we can find a corresponding disk about corresponding disk about zz00 such that all the such that all the points in the disk about points in the disk about zz00 are mapped into it. are mapped into it. That is, That is, any any neighborhood of neighborhood of ww00 contains all the contains all the values assumed by values assumed by ff in some full neighborhood in some full neighborhood of of zz00, except possibly , except possibly f(zf(z00))..

zz00 w

0

w = f(z)

z-plane w-planeu

v

x

y

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.

)()(lim 00

zfzfzz

Test for Continuity of Functions

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

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

it is true in a general metric space but not

in general topological space.

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 z0 provided 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:

i. f1(z)=z

ii. f2(z)=|z|

iii. f3(z)=z2

iv. f4(z)=

v. f5(z)=1/(z-2)

vi. f6(z)=ez/log(z)/z1./2/cos(z)

z

Test for Continuity of Functions

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

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

it is true in a general metric space but not

in general topological space.

Linear Map and Matrices

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

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

(x1, X2, --------- xn)nR

Every Vn is isomorphic to Rn

nn RV Its

Significance??

This isom

orphism is basis

dep

end

ent

Linear Map and Matrices

Every Vnm is isomorphic to

nmnmnm RRV

Its Significance??

This isom

orphism is basis

dep

end

ent

VnWm

k1v1+k2v2

k1L(v1)+k2L(v2))

L

L(k1v1+k2v2)

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

nmR

=

Example: Convergent Sequence

• Given , choose N=1/ , p=0

1/np n

10

Establish convergence by applying definitionNecessitates knowledge of p.

Convergent Sequence Properties

Cauchy Sequence

• A sequence in a metric space X such that for every , there is an integer N such that if ,m n N ,X n md p p

0 np

Example: Cauchy Sequence

• Given , choose N=2/, p=0

1 /nnp n

10

Establish convergence by applying definitionNo 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.

np np

k

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.

Home Task

Show that C/Z is completeWrite down diffrences between C and R2