Advanced mathematics for engineers with applications in stochastic processes / Aliakbar Montazer Haghighi, Jian-ao Lian, and Dimitar P. Mishev.

Format:

Book

Author/Creator:

Haghighi, Aliakbar Montazer.

Contributor:

Lian, Jian-ao.

Mishev, D. P. (Dimiter P.)

Series:

Mathematics research developments series.

Mathematics research developments

Language:

English

Subjects (All):

Functions of several complex variables.

Stochastic analysis.

Physical Description:

1 online resource (568 p.)

Edition:

Rev. ed.

Place of Publication:

New York : Nova Science Publishers, Inc., 2011, c2010.

Language Note:

English

Summary:

The contents of this work cover Fourier and wavelet analysis, Laplace transform, probability, statistics, difference and differential-difference equations, stochastic processes and their applications, and much more.

Contents:

Intro

ADVANCED MATHEMATICSFOR ENGINEERS WITH APPLICATIONSIN STOCHASTIC PROCESSES

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

CONTENTS

PREFACE

Chapter 1: INTRODUCTION

1.1. FUNCTIONS OF SEVERAL VARIABLES

Definition 1.1.1.

Example 1.1.1.

Definition 1.1.2.

Definition 1.1.3.

Definition 1.1.4.

Definition 1.1.5.

Example 1.1.2.

Definition 1.1.6.

Definition 1.1.7.

1.2. PARTIAL DERIVATIVES, GRADIENT, AND DIVERGENCE

Definition 1.2.1.

Theorem 1.2.1 (Clairaut's1 Theorem or Schwarz's2 Theorem)

Example 1.2.1.

Definition 1.2.2.

Example 1.2.3.

Definition 1.2.3.

Definition 1.2.4.

Definition 1.2.5.

Example 1.2.4.

Definition 1.2.6.

Definition 1.2.7.

Example 1.2.5.

Definition 1.2.8.

Theorem 1.2.2.

Example 1.2.6.

1.3. FUNCTIONS OF A COMPLEX VARIABLE

Definition 1.3.1.

1.4. POWER SERIES AND THEIR CONVERGENT BEHAVIOR

Definition 1.4.1.

Definition 1.4.2.

1.5. REAL-VALUED TAYLOR SERIES AND MACLAURIN SERIES

Definition 1.5.1.

Definition 1.5.2.

1.6. POWER SERIES REPRESENTATION OF ANALYTIC FUNCTIONS

1.6.1. Derivative and Analytic Functions

Definition 1.6.1.

Definition 1.6.2

Theorem 1.6.1 (Cauchy-Riemann10 Equations and Analytic Functions)

1.6.2. Line Integral in the Complex Plane

Definition 1.6.3.

Definition 1.6.4.

Definition 1.6.5.

Theorem 1.6.2.

1.6.3. Cauchy's Integral Theorem for Simply Connected Domains

Theorem 1.6.3 (Cauchy's Integral Theorem)

1.6.4. Cauchy's Integral Theorem for Multiple Connected Domains

Theorem 1.6.4. (Cauchy's Integral Theorem for Multiple ConnectedDomains)

1.6.5. Cauchy's Integral Formula

Theorem 1.6.5. (Cauchy's Integral Formula)

1.6.6. Cauchy's Integral Formula for Derivatives.

Theorem 1.6.6. (Cauchy's Integral Formula for Derivatives)

1.6.7. Taylor and Maclaurin Series of Complex-Valued Functions

Definition 1.6.6.

Definition 1.6.7.

Theorem 1.6.7. (Taylor Theorem)

Definition 1.6.8.

1.6.8. Taylor Polynomials and their Applications

Definition 1.6.9.

EXERCISES

1.1. Functions of Several Variables

1.2. Partial Derivatives, Gradient, and Divergence

1.3. Functions of a Complex Variable

1.4. Power Series and their Convergent Behavior

1.5. Real-Valued Taylor Series and Maclaurin Series

1.6. Power Series Representation of Analytic Functions

Chapter 2: FOURIER AND WAVELET ANALYSIS

2.1. VECTOR SPACES AND ORTHOGONALITY

Definition 2.1.1.

Definition 2.1.2.

Definition 2.1.3.

Definition 2.1.4.

Definition 2.1.5.

Definition 2.1.6.

Definition 2.1.7.

Definition 2.1.8.

Definition 2.1.9.

Definition 2.1.10.

Definition 2.1.11.

2.2. FOURIER SERIES AND ITS CONVERGENT BEHAVIOR

Definition 2.2.1.

Definition 2.2.2.

Definition 2.2.3.

Theorem 2.2.1. (Uniform Convergence)

Theorem 2.2.2. (Fourier Series of Piecewise Smooth Functions)

2.3. FOURIER COSINE AND SINE SERIESAND HALF-RANGE EXPANSIONS

Definition 2.3.1.

Definition 2.3.2.

2.4. FOURIER SERIES AND PDES

Definition 2.4.1.

2.5. FOURIER TRANSFORM AND INVERSE FOURIER TRANSFORM

Definition 2.5.1.

Definition 2.5.2.

2.6. PROPERTIES OF FOURIER TRANSFORMAND CONVOLUTION THEOREM

Definition 2.6.1.

2.7. DISCRETE FOURIER TRANSFORMAND FAST FOURIER TRANSFORM

Definition 2.7.1.

Definition 2.7.2.

Definition 2.7.3.

Definition 2.7.4.

2.8. CLASSICAL HAAR SCALING FUNCTION AND HAAR WAVELETS

Definition 2.8.1.

2.9. DAUBECHIES7 ORTHONORMALSCALING FUNCTIONS ANDWAVELETS

Definition 2.9.1.

Definition 2.9.2.

2.10.MULTIRESOLUTION ANALYSIS IN GENERAL

Definition 2.10.1.

2.11.WAVELET TRANSFORM AND INVERSE WAVELET TRANSFORM

Definition 2.11.1.

Definition 2.11.2.

2.12. OTHER WAVELETS

2.12.1. Compactly Supported Spline Wavelets

Definition 2.12.1.

Definition 2.12.2.

2.12.2. Morlet Wavelets

2.12.3. Gaussian Wavelets

2.12.4. Biorthogonal Wavelets

2.12.5. CDF 5/3 Wavelets

2.12.6. CDF 9/7 Wavelets

2.1. Vector Spaces and Orthogonality

2.2. Fourier Series and its Convergent Behavior

2.3. Fourier Cosine and Sine Series and Half-Range Expansions

2.4. Fourier Series and PDEs

2.5. Fourier Transform and Inverse Fourier Transform

2.6. Properties of Fourier Transform and Convolution Theorem

2.8. Classical Haar Scaling Function and Haar Wavelets

2.9. Daubechies Orthonormal Scaling Functions and Wavelets

2.12. Other Wavelets

Chapter 3: LAPLACE TRANSFORM

3.1. DEFINITIONS OF LAPLACE TRANSFORM ANDINVERSE LAPLACE TRANSFORM

Definition 3.1.1.

Theorem 3.1.1. (Existence of Laplace Transform)

3.2. FIRST SHIFTING THEOREM

Theorem 3.2.1. (First Shifting or s-Shifting Theorem)

3.3. LAPLACE TRANSFORM OF DERIVATIVES

Theorem 3.3.1. (Laplace Transform of First Order Derivative) .

Theorem 3.3.2. (Laplace Transform of High Order Derivatives)

3.4. SOLVING INITIAL-VALUE PROBLEMS BY LAPLACE TRANSFORM

3.5. HEAVISIDE FUNCTION AND SECOND SHIFTING THEOREM

Definition 3.5.1.

Theorem 3.5.1. (The Second Shifting or t-Shifting Theorem)

3.6. SOLVING INITIAL-VALUE PROBLEMSWITH DISCONTINUOUS INPUTS

3.7. SHORT IMPULSE AND DIRAC'S DELTA FUNCTIONS

3.8. SOLVING INITIAL-VALUE PROBLEMSWITH IMPULSE INPUTS

3.9. APPLICATION OF LAPLACE TRANSFORMTO ELECTRIC CIRCUITS

3.10. TABLE OF LAPLACE TRANSFORMS

3.1. Definitions of Laplace Transform and Inverse Laplace Transform

3.2. First Shifting Theorem

3.3. Laplace Transform of Derivatives.

3.4. Solving Initial-Value Problems by Laplace Transform

3.5. Heaviside Function and Second Shifting Theorem

3.6. Solving Initial-Value Problems with Discontinuous Inputs

3.8. Solving Initial-Value Problems with Impulse Inputs

3.9. Application of Laplace Transform to Electric Circuits

Chapter 4: PROBABILITY

4.1. INTRODUCTION

Definition 4.1.1.

Definition 4.1.2.

Definition 4.1.3.

Definition 4.1.4.

Definition 4.1.5.

Definition 4.1.6.

Definition 4.1.7.

Definition 4.1.8.

Definition 4.1.9.

4.2. COUNTING TECHNIQUES

Definition 4.2.1.

Rule 4.2.1. The Fundamental Principle of Counting

Definition 4.2.2.

Theorem 4.2.1.

Definition 4.2.3.

Definition 4.2.4.

Theorem 4.2.3.

4.3. TREE DIAGRAMS

4.4. CONDITIONAL PROBABILITY AND INDEPENDENCE

Definition 4.4.1.

Definition 4.4.2.

Theorem 4.4.1.

Definition 4.4.3.

4.5. THE LAW OF TOTAL PROBABILITY

Theorem 4.5.1. (The Multiplicative Law)

Theorem 4.5.2. (The Multiplicative Law)Let 1

Theorem 4.5.3. (The Law of Total Probability)

Theorem 4.5.4. (Bayes' Formula)

4.6. DISCRETE RANDOM VARIABLES

Definition 4.6.1.

Definition 4.6.2.

Definition 4.6.3.

4.7. DISCRETE PROBABILITY DISTRIBUTIONS

Definition 4.7.1.

Definition 4.7.2.

Definition 4.7.3.

Definition 4.7.4.

Definition 4.7.5.

Definition 4.7.6.

Definition 4.7.7.

Definition 4.7.8.

Definition 4.7.9.

Theorem 4.7.2.

4.8. RANDOM VECTORS

Definition 4.8.1.

Definition 4.8.2.

Definition 4.8.3.

Theorem 4.8.1. Multinomial Theorem

Definition 4.8.4.

4.9. CONDITIONAL DISTRIBUTION AND INDEPENDENCE

Theorem 4.9.1. (The Law of Total Probability)

Definition 4.9.1.

Definition 4.9.2.

Definition 4.9.3.

Theorem 4.9.2.

Theorem 4.9.3

Theorem 4.9.4.

4.10. DISCRETE MOMENTS

Definition 4.10.1.

Definition 4.10.2.

Theorem 4.10.1.

Theorem 4.10.2.

Theorem 4.10.3.

Definition 4.10.3.

Definition 4.10.4.

Definition 4.10.5.

Theorem 4.10.4.

Definition 4.10.6.

Theorem 4.10.5.

Definition 4.10.7.

Theorem 4.10.6.

Theorem 4.10.7.

Theorem 4.10.8.

Theorem 4.10.9.

Theorem 4.10.10.

Theorem 4.10.11.

Definition 4.10.8.

4.11. CONTINUOUS RANDOM VARIABLES AND DISTRIBUTIONS

Definition 4.11.1.

Definition 4.11.2.

Definition 4.11.3.

Definition 4.11.4.

Definition 4.11.5.

Definition 4.11.6.

Definition 4.11.7

Definition 4.11.8

Definition 4.11.9.

Definition 4.11.10

Definition 4.11.11.

Definition 4.11.12.

Definition 4.11.13.

Definition 4.11.14

Definition 4.11.15.

Definition 4.11.16

Remark 4.11.1.

4.12. CONTINUOUS RANDOM VECTOR

Definition 4.12.1.

Definition 4.12.2

4.13. FUNCTIONS OF A RANDOM VARIABLE

Definition 4.13.1.

Definition 4.13.2.

Theorem 4.13.1.

Definition 4.13.3.

Theorem 4.13.2.

Definition 4.13.4.

Theorem 4.13.3. Central Limit Theorem

4.1. Introduction

4.2. Counting Techniques

4.3. Tree Diagrams

4.4. Conditional Probability and Independence

4.5. The Law of Total Probability

4.6. Discrete Random Variables

4.7. Discrete Probability Distributions

4.8. Random Vectors

4.9. Conditional Distribution and Independence

4.10. Discrete Moments

4.11. Continuous Random Variables and Distributions

4.12. Continuous Random Vector

4.13. Functions of a Random Variable

Chapter 5: STATISTICS

PART ONE: DESCRIPTIVE STATISTICS

5.1. BASIC STATISTICAL CONCEPTS

Definition 5.1.1.

Definition 5.1.2.

5.1.1. Measures of Central Tendency

Definition 5.1.3.

Definition 5.1.4.

Definition 5.1.5.

Definition 5.1.6.

5.1.2. Organization of Data

Definition 5.1.7.

Definition 5.1.8.

Definition 5.1.9.

Notes:

Description based upon print version of record.

Includes bibliographical references (p. [535]-541) and index.

Description based on print version record.

ISBN:

1-62417-681-X

OCLC:

839304564