Complex Functions
Limit and Continuity
Mohammed Nasser
Acknowledgement:
Steve Cunningham
Relation between MM (ML) and Vector
space
Statistical
Concepts in Vector space
Concepts/Techniques
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.
Mathematical Concepts
Mathematical Concepts
Covariance
Variance
Z
Basis
F
f1
f3
f
0
Set of f1, f2, …, fn is linearly independent if α1 f1+ α2 f2 + …+ αn fn =
0 ,Holds only if each αi =0.
f 2 n linearly independent
Finite dimensional if there exist maximum
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
Basis ( Continued)
Mathematical Concepts
Covariance
Variance
Some Vector Concepts
• Dot product =
scalar
x y  x 1 x 2

T
y 
1
 
y 2 

... x n

  
 
y n 
x y  x1 x2
T
n
x y
i i
i 1
• Length of a vector
|| x || = (x1
2+
x2
2+
x3
 y1 
3
 
x3  y2  x1 y1  x2 y2  x3 y3   xi yi
 
i 1
 y3 
Right-angle triangle
x2
2 )1/2
Inner product of a vector
with itself = (vector
length)2
xT x =x12+ x22 +x32 = (|| x
||)2
x2
Pythagoras’ theorem
|| x || =
(x12+ x22)1/2
||x||
x1
x1
Some Vector Concepts
• Angle between two
vectors
sin  
y2
sin  
x2
y
x
cos  
y1
cos  
x1
||x||

||y||

y

y1
x
cos   cos(    )  cos  cos   sin  sin  
y 1x 1  y 2x 2
x
x y
T
cos  
y2
x y
x y
x
x y  x y cos 
T
Orthogonal vectors:
xT y = 0
=/
2
y
Dot Product in Rn
T
x y  x 1 x 2

f ;R
n
R
n
y 
1
 
y2 

... x n 

  
 
y n 
n
x y
i i
i 1
 R
XTy=yTx
XTX=0 ↔ X=0
(aX+bZ)Ty=aXTY+bZTY
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
Show that
1) <x,y>=<y,x>
R is itself
2) <x,x>.≥0, and <x, x>=0↔ x=0
an inner
product
3) <x+z,y>=<x,y>+<z,y>
space with
4) <kx,y>=k<x,y>
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
Show that
1) <x,y>=<y,x>
C/Z is itself
2) <x,x>.≥0, and <x, x>=0↔ x=0
an inner
3) <x+z,y>=<x,y>+<z,y>
product
4) <kx,y>=k<x,y>
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 3:<cu, v>= <c(x1, x2), (y1, y2)>
=< (cx1, cx2), (y1, y2) >
= cx1y1 + 4cx2y2 = c(x1y1 + 4x2y2)
= c<u, v>
2
2
x1  4 x 2  0
Axiom 4: <u, u>= <(x1, x2), (x1, x2)>=
2
2
Further, x1  4 x 2  0 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.
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.
u





a
c
e
b 

,
v



g
d 

f
h





Prove that the following function is an inner product on M22.
<u, v>= ae + bf + cg + dh
Determine the inner product of the matrices
2

0
 3
 and
1 
5

9
2
.
0
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>
2
 0
 3 5
,

1   9
2
0 
 ( 2  5 )  (  3  2 )  ( 0  9 )  (1  0 )  4
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
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
cos  
uv
(??)
u v
Definition
Let V be an inner product space. The angle  between two
nonzero vectors u and v in V is given by
cos  
u, v
u v
Normed spaces
Define the notion of the size of f, an
element in F , a vector space vector sce
• Norm || f ||, || ||: F→[0,∞)
1)||f||=0↔f=0
2) ||kf||=|k|||f||
3) ||f|+||g|| <=||f||+||g||
f  0,
f 0 f 0
Both R and Z are normed spaces are spaces with | |
f1 
n
R and
f 2  f1  f 2
n m
Both
are normed spaces are spaces
R
with Euclidean norm,|| || d  f 1 , f 2   f 1  f 2
Norm of a Vector
The norm of a vector in Rn can be expressed in terms of the dot
product as follows
( x1 , x 2 ,  , x n ) 

( x1    x n )
2
2
( x1 , x 2 ,  , x n )  ( x1 , x 2 ,  , x n )
Generalize this definition:
The norms in general vector space do not necessary have geometric
interpretations, but are often important in numerical work.
Definition
Let V be an inner product space. The norm of a vector v is
denoted ||v|| and it defined by
v 
v, v
Example
Consider the vector space M22 of 2  2 matrices. Let u and v
defined as follows be arbitrary 2  2 matrices.
u





a
c
e
b 

,
v



g
d 

f
h





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 || u ||
u, u 
a b c d
2
2
2
2
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
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.
Definition
Let V be an inner product space with vector norm defined by
v 
v, v
The distance between two vectors (points) u and v is defined
d(u,v) and is defined by
d (u , v )  u  v
(
u  v, u  v )
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 space
nonnegative function d:F χF→R
F defining
f2
F
f1
f1
d ( f1 , f 2 )
f2
1.d(f1,, f2)=d(f2,, f1), 2) d(f1,, f2)=0
3, d(f1,, f2)≤
d(f1,, f3) + d(f3,, f2)
if and only if 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
We
limit and continuity of metric space valued functions
on metric space.
Define
Rate • Completeness – no holes in the space
• We want to look at spaces that are very similar to
Of
Euclidean space
Change??
Can we talk about best approximations?
Can
Yes if
Can we get the best from data?
Mathematical Concepts
Covariance
Variance
Sequence
• Definition. A sequence of complex numbers,
denoted
k
zn1 , 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

zn1, 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
Zn
Z0
|zn-z0|=rn
Where rn=
, xn
x0,,, yn
Z0
0
( x n  x 0 )  (y n  y 0 )
2
2
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 zn1
is said to have the limit z0 , or to converge to z0 ,
if for any e > 0, there exists an integer N such
that |zn – z0| < e for all n > N. We denote this by
lim zn  z0 or
n
zn  z0 as n  .
• 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 .
Geometric Meaning of Zn
Z0
zN+2
z0
zN+1
zn tends to z0 in any linear or curvilinear way.
Example: Convergent Sequence
p n  1 / n or d ( s n , s )  1 / n or z n  z  1 / n
• Given e, choose N=1/ e, p=0
0
1
Establish convergence by applying definition
Necessitates knowledge of p.
Cauchy Sequence
A sequence  s n  in a metric space X such that for
every e  0 , there is an integer N such that if
m , n  N , d X  sn , sm   e
A sequence { z n } in a complex field C such that for
every e  0 , there is an integer N such that if
m, n  N ,
zn  zm  e
Example Cauchy Sequence
pn  1 / n
n
• Given e, choose N=2/e, p=0
0
1
Establish convergence by applying definition
No need to know p
Theorems and Exercises
Theorem. Show that zn=xn+iyn
and only if xn x0, yn y0 .
z0=x0+iy0 if
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 of
object under continuous deformations
e
d
For all e > 0, there exists d > 0
: |y-x| < d ) |f(y)-f(x)| < e
Topology
Topology studies the invariant properties of
object under continuous deformations
S
f-1(S)
For all S open, f-1(S) is open
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 e
> 0 there exists a positive real number d > 0
such that |f(z) – w0|< e whenever 0<|z - z0|< d.
Limits: Interpretation
We can interpret this to mean that if we observe points w within a radius d of w0,
we can find a corresponding disk about z0 such that all the points in the disk about
z0 are 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)
d z
0
z-plane
e
w0
x
w-plane
u
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
lim f ( z)  f ( z0 )
– zz0
• 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
For all sn
s
<S2,d2> is continuous
f(sn)
f(s)
it is true in a general metric space but not
in general topological space.
f: <S1,d1>
at s in S1.
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
•
i.
ii.
iii.
iv.
v.
vi.
Find domain and range of the following
functions and check their continuity:
f1(z)=z
f2(z)=|z|
f3(z)=z2
f4(z)= z
f5(z)=1/(z-2)
f6(z)=ez/log(z)/z1./2/cos(z)
Test for Continuity of Functions
For all sn
s
<S2,d2> is continuous
f(sn)
f(s)
it is true in a general metric space but not
in general topological space.
f: <S1,d1>
at s in S1.
Linear Map and Matrices
v=x1v1+x2v2+--------+xnvn
(x1,
X2,
xn)
---------
R
n
Every Vn is isomorphic to Rn
Vn  R
n
Its
Significance??
This isomorphism is basis
dependent
Vn is a finite-dimensional vector space. Let
v belongs to Vn
Linear Map and Matrices
L
L(k1v1+k2v2)
Wm
=
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
V nm  R
mn
 R
R
mn
nm
Its
Significance??
This isomorphism is basis
dependent
Vn
Example: Convergent Sequence
pn  1 / n
• Given e, choose N=1/ e, p=0
0
1
Establish convergence by applying definition
Necessitates knowledge of p.
Convergent Sequence Properties
Cauchy Sequence
• A sequence  p  in a metric space X such
that for every e  0 , there is an integer N
such that if m , n  N d  p , p   e
n
X
n
m
Example: Cauchy Sequence
pn  1 / n
n
• Given e, choose N=2/e, p=0
0
1
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  p 
is a Cauchy sequence in X, then  p 
converges to some point of X.
• (c) In k , every Cauchy sequence
converges.
n
n
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 complete
Write down diffrences between C and R2