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