math for cs

8/12/2019 Math for CS

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POSTGRADUATE LECTURE

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Fundamentals of Algorithm

PART I CHAPTERMATHEMATICAL BASICS

for:MSc students in Computer Science

14-Mar-14 N.P. Khu


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CONTENTS

1. Complex Numbers

2. Complex Power Series3. Generating Functions

4. Harmonic &Bernoui l l i Number

5. Asymptotic Behavior

1.0

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1. Definitions and Properties

a. F ield of Complex Numbers

How we can find a root of the equation ?

What abou t a new number i that ?

Complex numbers, can be defined as pairs of real numbers

equipped with the addition: +

and the multiplication: .

form a field, in which is an Abel groupwith

element uni t , and is also an Abel groupwith

its unit element, and . distributedto +operation.

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b. Representations

Facts:

So, can be though of as real numbers and

we mean a combination of with coefficients .

Let us denote byi , thencan be written as: .

From this, and

are called the real

and imaginary part, respectively.

Identity: is reads as : !

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c. Modulus and Argument

Modulusof absolute value of is defined as:

Argumentof is a number such that :

Distancebetween

is :

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d. Euler formula

Let

it is obtained:

Euler formula

then:

Conjugateof is . From this, we have:

for a triangle inequality:

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e. Some Properties

Some basic properties of the conjugate, for any :

Identities satisfying for all :

-a rewrite of the triangle inequality,

-this is a reverse triangle inequality

that can be proved as an exercise,

-this follows by induction.

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2. Limits - Continuous

a. Limits

Open disk with center a, radius r :

closed disk and boundary:

A complex function:e.g. :

A limit: is defined as

Note: does not exist ! Because, on the real and imaginary

while

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2. Limits - Continuous

b. Continuity

If and exist, then:

In case of then the function is continuous at .

A function is continuouson a domain if it is at every .If is continuous at and then it is obtained:

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a. Derivatives

Suppose is a complex function and is an interior pointof . The derivativeof at is defined as:

or

If this limit exists. Then is called dif ferentiable at .. If thefunction is differentiable all points in an open disk centered at

then is holomorphicor analyticat .

The function is holomorphic on the open set if it is

differentiable at every point . Functions which are

differentiable in the whole complex plane are called entire.

is entire, , it is holomorphic for any

is nowheredifferentiable due to

does not existN.P. Khu

3. Derivatives

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a. Derivatives

Suppose and are differentiable at , and ,and is differentiable at :

and are open sets in , is a bijection,is the inversefunction of , . . If f is differentiable at ,

, and is continuous at then is differentiable at

with:

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3. Derivatives

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b. Cauchy-Riemann Equations

If is differentiable at , then:

[Cauchy-Riemann]

Suppose is such that the partial derivatives and exist in an

open disk centered at and are continuous at . If these partialderivatives satisfy the C-R equation then is differentiable at .

Based on these above cases, the derivative at is given by:

If we write then and

, in this case, Eq. C-R becomes:

It is obtained: (Laplace Equation).12N.P. Khu

3. Derivatives

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c. Exponential Function

Complex exponential functionis defined for as

Some properties follow from Euler formulaSlide no. 6, satisfying

for all are:

The 3rd identity is a very special one and has no counterpart for

the real exponential function: periodicity with period 2.For then .

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4. Special Functions

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d. Trigonometr ic Function

Complex sine, cosineand tangent, cotangent are definedas:

and

and

Properties: for all it is obtained:

Warning:

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e. Logar ithmic Function

An inverse function to the complex exponential function is

the complex logari thm , such that:

that is and

. The logarithm function would be .

Given a region , any continuous function that

satisfies is a branch of the logarithm (on ). Let

denote that argument of which is in . Then the

principal logari thmis defined as .

Suppose is a branch of the logarithm. Then is differentiable

wherever it is continuous

and:

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5. Curvilinear Integration

a. I ntegration on Smooth Curve

Let be a smooth curveparametrized, andis a complex function which is continuous on , we define integral

of on as

This definition can be extended to piecewise smooth curves.

Example: Let over parabol with

from to . A parametrization of this curve

is , and . Thus:

whence

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1. Sequences

a. Convergence-Divergence

Suppose is a sequence and such that for all , thereis an integer such that for all : , in such a case

the sequence is convergent and is its limit:

If no such exists, the sequence is divergent.

Example 1: We have: . For any , let ,

then

Example 2: Sequence diverges. Let and :

if : for any and , let then .

if : doing the same, letting also has .

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1. Sequences

b. Related Properties

Let be convergent sequences and , we have:

In(c)we have to make sure we do not divide by zero.

Monotone Property: Any bounded monotone sequence converges.

Archimedean Property: If is any real number, there is an

integer which is greater than .

Exponenti als beat polynomials: for any polynomial and

any , it is obtained .

Factorials beat exponentials: if , then .

Note these last properties also works for !

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2. Series

a. Definitions

A series is a whose members are orpartial sums of the series, where is the sequence of terms.

A series converges to the limit by definition if :

for any there exists ,

or so that for all .

Example 1: For the telescoping series, we have:

Example 2: Harmonic series diverges, due to this fact:

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2. Series

b. Absolute Convergence

Series converges absolutelyif .

If a series converges absolutely then it converges.

Example: The alternating harmonic series: converges,

but not absolutely.

In fact,

It is verified , , the general term satisfies:

The series converges by comparison Example 1 in previous slide.

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2 CO O S S


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2. Series

c. Sequences and Ser ies of Functions

Sequence of functions convergesat if the sequence ofcomplex numbers converges. If a sequence of functions,

converges at all in then converges pointwiseon .

Suppose and are functions defined on . If for all

there is an such that for all and for all we havethen converges uni formlyin to .

Let be a sequence of continuous functions on converging

uniformly to on . Then is continuous on .

Suppose are continuous on the region , , for alland converges. Then converges absolutely

and uniformly in , Weierstr ass M-Test .

Consider on converges pointwise to 0or 1.

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2 COMPLEX POWER SERIES


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3. Power Series

a. Definition

A power series centered at is a series of functions of the form:

+ The geometr ic series converges absolutely for to

the function .+ The convergence is uniformon any set of the form

for any .

This is straightforward by using Weierstrass M-Test and letting:

with .

Finally, for all .

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3. Power Series

b. Region of Convergence

Let an open disk with radius centred at 0,

and corresponding closed disk: .

Any power series has a radius of convergence .

By this, is a nonnegative real number, or , satisfying:

a. If :, converges absolutelyand uniformlyon

the closed disk of radius centered at .

b. If : the sequence of terms is unbounded,

so does not converge, or divergences.

Open disk in which the power series converges absolutely isthe region of convergence. If then is the entire

complex plane, and if then is the empty, e.g. :

Ratio Test shows that, has a radius of convergence .

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3. Power Series

b. Region of Convergence

Suppose has radius of convergence , thenis differentiable, holomorphic, analyticin .

has the same radius of convergence .

Example: Consider and , it is obtained

From and ,

this means that for some constant .

Evaluating at , we see that so as desired !

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3 GENERATING FUNCTIONS


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1. Definition

a. Definition

Consider a sequence of real numbers, thegenerating function of this sequence is defined on as:

for those values of s for which the sum

there exists a radius of convergence such that

the series convergent if .

Some of useful generating functions:Polynomial: is entieron .

Exponential: is also entieron , in which

Triginometrics:

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b. Harmonic Numbers

It is obtained: converges absolutely if .

Taking its integration, term by term, we have:

Hence,

By multiplying the above series of and , we have:

where

Hence, is the generating functionof , the so-called

harmonic sequenceor harmonic numbersw.r.t n.

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1. Definition



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2. Generating Probabilities

a. Probabil i ty Generating Functions

Consider a count r.v. , i.e. a discrete r.v. taking non-negativevalues, and , probabil i ty generating

functionis:

From , the series converges absolutely for , e.g.

: Binomialr.v. ,

: Poissonr.v. ,

: Negative Binomialr.v. has generating function:

if , .

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b. Uniqueness

If and have probability generating function andrespectively, then it is obtained :

iff

i.e. iff and have the same probability distribution.

Let be a count r.v. and the rthderivative of itsat . Then

So, and

That is :

e.g. , then , ,

and finally,

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3. An Application

a. Solving Recurrence Relation

Matching Problem: it is faced with the recurrence relation

for the probability that no matches occur in an n-card matching

problem. Multiply through by and sum over:

Let we can write , provided that

with condition :

Expand and extract the coefficient of , then for :

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4 BERNOUILLI NUMBER


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1. Bernouilli Polynomial

a. Bernouil l i Numbers

The generating function for the Bernoull i numbersis

From:

Then:

Replacing and using , it is obtained:

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Radius of convergence: the generating function is singularwhenever , except for . Thus the closest singularities to

the real axis occur at 2i , so that the radius of convergence

is 2, and :

It is inferred that the Bernoulli numbers grow rapidly with n .

We can not deduce the sign or overall constant from this analysis:

The true asymptotic behavior of is :

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Bernouilli numbers compared with its above approximation:

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b. Bernouil li Polynomials

The Bernoulli polynomials are defined by the generating function

where

Properties of these polynomials can be deduced as follows,

Bernoulli polynomials at zero are equal to the Bernoulli numbers

with

and

by comparing corresponding terms in the generating function

expansion, we find

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2. Properties


Dierentiate the generating function w.r.t its 2ndargument:and

Equating coecients of we have: .By direct power series expansion of the generating function, the 1st

few Bernoulli polynomials can be read o:

So, and some others also be found, as follows:

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2. Properties


Direct power series expansion of the generating function we have:

from which we read o:

By keeping two more terms in the expansion we find

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4.BERNOUILLI NUMBER

5 ASYMPTOTIC BEHAVIORS


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1. Stirling Formula

a. Theorem:

Proof :

using the following steps:

1. Verify that:

2. Show that: is a bounded, monotone decreasing sequence.

3. Conclude that: , for some .

4. Conclude that: for some positive

real number .

5. Show that: using the following formula, named Wallis:

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2. Approximations

b. Some Approximations

Euler constant:

whence

By using 6 terms:

Stirling constant: .Using complex integral : , in general, it is obtained:

Replacing :

hence .

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Home works:1. file BT-x, x = 1, 2, 3, 4 in:

MScMath-Excer-1.zip

Assignments:1. read extensions in zip-file

MScMath-Assig-1.zip

Assignment

math for cs

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