# Advanced Engineering Mathematics

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## Book Description

Based on the authors' three decades of teaching experience, Advanced Engineering Mathematics presents the fundamentals and theoretical concepts of the subject in an intelligible and easy-to-understand style. The carefully planned chapters make this book an effective tool for teaching the application of mathematics to engineering and scientific problems.

The book provides graded sets of examples and exercises that help students understand the concepts and an Answers and Hints section for difficult problems. The exercises emphasize theoretical concepts and provide the practice students need to gain the confidence to apply them. Advanced Engineering Mathematics offers a logical and lucid presentation of both theory and problem solving techniques that ensures students will not ett lost in unnecessary details.

## Table of Contents

FUNCTIONS OF A REAL VARIABLE

Introduction

Limits, Continuity, and Differentiability

Application of Derivatives and Taylor's Series

Integration and its Applications

Improper Integrals

FUNCTIONS OF SEVERAL REAL VARIABLES

Introduction

Functions of Two Varaibles

Partial Derivatives

Higher Order Partial Derivatives

Maximum and Minimum Values of Functions

Multiple Integrals

MATRICES AND EIGENVALUE PROBLEMS

Introduction

Matrices

Vector Spaces

Solution of General Linear System of Equations

Eigenvalue Problems

Answers and Hints

ORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDER

Introduction

Formation of Differential Equations

Solution of a Differential Equation

Initial and Boundary Value Problems

Solution of Equations in Separable Form

Exact First Order Differential Equations

Linear First Order Equations

Some Special First Order Equations

Orthogonal Trajectories of a Given Family of Curves

Existence and Uniqueness of Solutions

Answers and Hints

LINEAR DIFFERENTIAL EQUATIONS

Introduction

Solution of Linear Differential Equations

Methods for Solution of Linear Equations

Solution of Non-Homogenous Linear Equations

Operator Methods for Finding Particular Integrals

Simultaneous Linear Equations

Answers and Hints

SERIES SOLUTION OF DIFFERENTIAL EQUATIONS

Introduction

Ordinary and Singular Points of an Equation

Power Series Solution

Series Solution about a Regular Singular Point: Frobenius Method

Answers and Hints

LEGENDRE POLYNOMIALS, BESSEL FUNCTIONS AND STURM-LIOUVILLE PROBLEM

Introduction

Legendre Differential Equation and Legendre Polynomials

Bessel's Differential Equation and Bessel's Functions

Sturm-Liouville Problem

Answers and Hints

LAPLACE TRANSFORMATION

Introduction

Basic Theory of Laplace Transforms

Laplace Transform Solution of Initial Value Problems

Translation Theorems (Shifting Theorems)

Laplace Transform of Dirac-delta Function and More Properties of Laplace Transforms

Laplace Transform of Periodic Functions

Laplace Transform Method for the Solution of Some Partial Differential Equations

Answers and Hints

FOURIER SERIES, FOURIER INTEGRALS AND FOURIER TRANSFORMS

Introduction

Fourier Series

Fourier Half-Range Series

Fourier Integrals

Application of Fourier Series: Separation of Variables Solution of Linear Partial Differential Equations

Fourier Transforms

Answers and Hints

FUNCTIONS OF A COMPLEX VARIABLE: ANALYTIC FUNCTIONS

Introduction

Complex Number System

Sets of Points in the Complex Plane

Functions of a Complex Variable

Elementary Functions

Limits and Continuity

Differentiability and Analyticity

Harmonic Functions

Answers and Hints

INTEGRATION OF COMPLEX FUNCTIONS

Introduction

Definite Integrals

Cauchy Integral Theorem

Chauchy Integral Formula

Cauchy Integral Formula for Derivatives

Answers and Hints

POWER SERIES, TAYLOR AND LAURENT SERIES

Introduction

Infinite Sequences

Infinite Series

Power Series

Taylor Series

Laurent Series

Answers and Hints

ZEROS, SINGULARITIES AND RESIDUES

Introduction

Zeros and Singularities of Complex Functions

Residues

Evaluation of Contour Integrals Using Residues

Evaluation of Real Integrals Using Residues

Answers and Hints

BILINEAR TRANSFORMATION AND CONFORMAL MAPPING

Introduction

Linear and Inverse Transformations

Bilinear Transformations

Conformal Mapping

Answers and Hints

VECTOR DIFFERENTIAL AND INTEGRAL CALCULUS

Introduction

Parametric Representatives, Continuity and Differentiability of Vector Functions

Gradient of a Scalar Field and Directional Derivative

Divergence and Curl of a Vector Field

Line Integrals and Green's Theorem

Surface Area and Surface Integrals

Divergence Theorem of Gauss and Stoke's Theorem

Answers and Hints