Engineering Mathematics: Volume - 2

Cover
About the Author
Contents
Preface
Chapter 1: Eigenvalues and Eigenvectors
	1.1 Introduction
		1.1.1 Matrix Polynomial
	1.2 Linear Transformation
	1.3 Characteristic Value Problem
		1.3.1 Characteristic Equation of Matrix A
		1.3.2 Spectrum of A
		1.3.3 Procedure for Finding Eigenvalues and Eigenvectors
	Exercise 1.1
	1.4 Properties of Eigenvalues and Eigenvectors
		1.4.1 Characteristic Polynomial Pn
	1.5 Cayley–Hamilton Theorem
		1.5.1 Inverse of a Matrix by Cayley–Hamilton Theorem
	Exercise 1.2
	1.6 Reduction of a Square Matrix to Diagonal Form
		1.6.1 Diagonalisation—Powers of a Square Matrix A
		1.6.2 Modal Matrix and Spectral Matrix of a Square Matrix A
		1.6.3 Similarity of Matrices
		1.6.4 Diagonalisation—Conditions for Diagonalisability of a Matrix A
		1.6.5 Orthogonalisation of a Symmetric Matrix
	1.7 Powers of a Square Matrix A— Finding of Modal Matrix P and Inverse Matrix A−1
		1.7.1 Solution over the Complex Field—Eigenvectors of a Real Matrix Over Complex Field
	Exercise 1.3
Chapter 2: Quadratic Forms
	2.1 Introduction
	2.2 Quadratic Forms
		2.2.1 Quadratic Form: Definition
	2.3 Canonical Form (or) Sum of the Squares Form
		2.3.1 Index and Signature of a Real Quadratic Form
	2.4 Nature of Real Quadratic Forms
		2.4.1 Positive Definite
		2.4.2 Negative Definite
		2.4.3 Positive Semi-Definite
		2.4.4 Negative Semi-Definite
		2.4.5 Indefinite
	2.5 Reduction of a Quadratic Form to Canonical Form
	2.6 Sylvestor’s Law of Inertia
	2.7 Methods of Reduction of a Quadratic Form to a Canonical Form
		2.7.1 Diagonalisation (by Simultaneous Application of Row and Column Transformations)
		2.7.2 Orthogonalisation
		2.7.3 Lagrange’s Method of Reduction (Completing Squares)
	2.8 Singular Value Decomposition of a Matrix
	Exercise 2.1
Chapter 3: Solution of Algebraic and Transcendental Equations
	3.1 Introduction to Numerical Methods
	3.2 Errors and their Computation
		3.2.1 Exact and Approximate Numbers
		3.2.2 Significant Digits
		3.2.3 Loss of Significant Digits
		3.2.4 Rounding off
		3.2.5 Rules for Rounding off
		3.2.6 Absolute, Relative and Percentage Errors
	3.3 Formulas for Errors
		3.3.1 Relative Error
		3.3.2 Error Bound for ã
		3.3.3 Error Propagation
		3.3.4 Error in Rounding
		3.3.5 Programming Errors
		3.3.6 Errors of Numerical Results
	3.4 Mathematical Pre-requisites
	3.5 Solution of Algebraic and Transcendental Equations
		3.3.1 Introduction
		3.3.2 Zero or Root of a Function
	3.6 Direct Methods of Solution
		3.6.1 Descartes’ 4 Rule of Signs
	3.7 Numerical Methods of Solution of Equations of the Form f(x) = 0
		3.7.1 Fixed Point Iteration (Successive Approximation) Method
		3.7.2 Bolzano’s5 (Bisection or Interval-Halving) Method
		3.7.3 Newton–Raphson6 Method
		3.7.4 Secants Method (or Chords Method)
		3.7.5 Method of False Position (Regula Falsi)
	Exercise 3.1
Chapter 4: Interpolation
	4.1 Introduction
		4.1.1 Formula for Errors in Polynomial Interpolation
	4.2 Interpolation with Equal Intervals
		4.2.1 Finite Differences
		4.2.2 Forward (Advancing) Difference Operator
		4.2.3 Properties Satisfied by
		4.2.4 Backward Difference Operator
	4.3 Symbolic Relations and Separation of Symbols
		4.3.1 Factorial Function
		4.3.2 The Enlargement or Displacement or Shift Operator E
		4.3.3 The Relations between
	Exercise 4.1
	4.4 Interpolation
		4.4.1 The Differential Operator D: Relation between
	4.5 Interpolation Formulas for Equal Intervals
		4.5.1 Newton1–Gregory2 Forward Interpolation Formula
		4.5.2 Newton–Gregory Backward Interpolation Formula
	Exercise 4.2
	4.6 Interpolation with Unequal Intervals
		4.4.1 Divided Differences
		4.4.2 Divided Differences: Notation
	4.7 Properties Satisfied by
		4.7.1 Linearity Property
		4.7.2 Symmetrical Property
		4.7.3 Vanishing of (n + 1) Divided Differences
		4.7.4 Special Case: Equally spaced Divided Differences
	4.8 Divided Difference Interpolation Formula
		4.8.1 Newton’s Divided Difference Formula
		4.8.2 Sheppard’s3 Zig-Zag Rule
		4.8.3 Lagrange’s4 Formula for Unequal Intervals
	4.9 Inverse Interpolation Using Lagrange’s Interpolation Formula
	4 .10 Central Difference Formulas
		4.10.1 Gauss’s5 Interpolation Formulae
		4.10.2 Stirling’s6 Formula
		4.10.3 Bessel7 Formula
	Exercise 4.3
Chapter 5: Curve Fitting
	5.1 Introduction
		5.1.1 Curve Fitting: Method of Least Squares
		5.1.2 Some Standard Approximating Curves
	5.2 Curve Fitting by the Method of Least Squares
		5.2.1 Least Squares Straight Line Fit or Linear Regression
		5.2.2 Least Squares Parabolic (Quadratic) Curve
		5.2.3 Nonlinear Curves
	5.3 Curvilinear (or Nonlinear) Regression
		5.3.1 Polynomial Regression
		5.3.2 Transcendental Curves
	5.4 Curve Fitting by a Sum of Exponentials
	5.5 Weighted Least Squares Approximation
		5.5.1 Linear Weighted Least Squares Approximation
		5.5.2 Nonlinear Weighted Least Squares Approximation
	Exercise 5.1
Chapter 6: Numerical Differentiation and Integration
	6.1 Introduction
		6.1.1 Numerical Differentiation
		6.1.2 Numerical Differentiation by Newton’s Forward Interpolation Formula
		6.1.3 Numerical Differentiation by Newton’s Backward Interpolation Formula
		6.1.4 Numerical Differentiation by Stirling’s Formula
	6.2 Errors in Numerical Differentiation
		6.2.1 Truncation Error
		6.2.2 Rounding Error
	6.3 Maximum and Minimum Values of a Tabulated Function
	Exercise 6.1
	6.4 Numerical Integration: Introduction
		6.4.1 Newton−Cotes Quadrature Formula
		6.4.2 Trapezoidal Rule
		6.4.3 Simpson’s1 13 Rule
		6.4.4 Simpson’s 38 Rule
		6.4.5 Boole’s2 Rule
		6.4.6 Weddle’s3 Rule
	Exercise 6.2
	6.5 Cubic Splines
		6.5.1 Interpolation by Spline Functions
		6.5.2 Numerical Differentiation: Cubic Spline Method
		6.5.3 Numerical Integration: Cubic Spline Method
	6.6 Gaussian Integration
	Exercise 6.3
Chapter 7: Numerical Solution of Ordinary Differential Equations
	7.1 Introduction
		7.1.1 Ordinary Differential Equation
		7.1.2 Initial or Boundary Value Problem
	7.2 Methods of Solution
		7.2.1 Method 1: Taylor’s Series Method
		7.2.2 Method 2: Picard’s Method of Successive Approximations
		7.2.3 Method 3: Euler’s Method
		7.2.4 Method 4: Euler’s Modified Method or Heun’s Method
		7.2.5 Method 5: Runge1–Kutta2 Methods
	7.3 Predictor–Corrector Methods
		7.3.1 Introduction
		7.3.2 Milne–Simpson’s Method
		7.3.3 Adams–Bashforth–Moulton (ABM) Method
	Exercise 7.1
Chapter 8: Fourier Series
	8.1 Introduction
	8.2 Periodic Functions, Properties
		8.2.1 Properties of Periodic Functions
	8.3 Classifiable Functions—Even and Odd Functions
	8.4 Fourier Series, Fourier Coefficients and Euler’s Formulae in (α , α +2π)
		8.4.1 Determination of Fourier Coefficients
	8.5 Dirichlet’s Conditions for Fourier Series Expansion of a Function
	8.6 Fourier Series Expansions: Even/Odd Functions
		8.6.1 Fourier Series of Odd and Even Functions in the Interval (−l, l )
		8.6.2 Fourier Series of Odd and Even Functions in the Interval (−π, π)
	8.7 Simply-Defined and Multiply-(Piecewise) Defined Functions
	Exercise 8.1
	8.8 Change of Interval: Fourier Series in Interval (α, α + 2l)
		8.8.1 Fourier Series in any Arbitrary Interval (a, b)
	Exercise 8.2
	8.9 Fourier Series Expansions of Even and Odd Functions in (−l, l )
	Exercise 8.3
	8.10 Half-Range Fourier Sine/Cosine Series: Odd and Even Periodic Continuations
	Exercise 8.4
	8.11 Root Mean Square (RMS) Value of a Function
		8.11.1 Parseval’s Formula
		8.11.2 By Parseval’s Related Formulas
	Exercise 8.5
Chapter 9: Fourier Integral Transforms
	9.1 Introduction
	9.2 Integral Transforms
		9.2.1 Laplace Transform
		9.2.2 Fourier Transform
	9.3 Fourier Integral Theorem
		9.3.1 Fourier Sine and Cosine Integrals (FSI’s and FCI’s)
	9.4 Fourier Integral in Complex Form
		9.4.1 Fourier Integral Representation of a Function
	9.5 Fourier Transform of f (x)
		9.5.1 Fourier Sine Transform (FST) and Fourier Cosine Transform (FCT)
	9.6 Finite Fourier Sine Transform and Finite Fourier Cosine Transform (FFCT)
		9.6.1 FT, FST and FCT Alternative definitions
	9.7 Convolution Theorem for Fourier Transforms
		9.7.1 Convolution
		9.7.2 Convolution Theorem
		9.7.3 Relation between Laplace and Fourier Transforms
	9.8 Properties of Fourier Transform
		9.8.1. Linearity Property
		9.8.2. Change of Scale Property or Damping Rule
		9.8.3. Shifting Property
		9.8.4. Modulation Theorem
	Exercise 9.1
	9.9 Parseval’s Identity for Fourier Transforms
	9.10 Parseval’s Identities for Fourier Sine and Cosine Transforms
	Exercise 9.2
Chapter 10: Partial Differential Equations
	10.1 Introduction
	10.2 Order, Linearity and Homogeneity of a Partial Differential Equation
		10.2.1 Order
		10.2.2 Linearity
		10.2.3 Homogeneity
	10.3 Origin of Partial Differential Equation
	10.4 Formation of Partial Differential Equation by Elimination of Two Arbitrary Constants
	Exercise 10.1
	10.5 Formation of Partial Differential Equations by Elimination of Arbitrary Functions
	Exercise 10.2
	10.6 Classification of First-Order Partial Differential Equations
		10.6.1 Linear Equation
		10.6.2 Semi-Linear Equation
		10.6.3 Quasi-Linear Equation
		10.6.4 Nonlinear Equation
	10.7 Classification of Solutions of First-Order Partial Differential Equation
		10.7.1 Complete Integral
		10.7.2 General Integral
		10.7.3 Particular Integral
		10.7.4 Singular Integral
	10.8 Equations Solvable by Direct Integration
	Exercise 10.3
	10.9 Quasi-Linear Equations of First Order
	10.10 Solution of Linear, Semi-Linear and Quasi-Linear Equations
		10.10.1 All the Variables are Separable
		10.10.2 Two Variables are Separable
		10.10.3 Method of Multipliers
	Exercise 10.4
	10.11 Nonlinear Equations of First Order
	Exercise 10.5
	10.12 Euler’s Method of Separation of Variables
	Exercise 10.6
	10.13 Classification of Second-Order Partial Differential Equations
		10.13.1 Introduction
		10.13.2 Classification of Equations
		10.13.3 Initial and Boundary Value Problems and their Solution
		10.13.4 Solution of One-dimensional Heat Equation (or diffusion equation)
		10.13.5 One-dimensional Wave Equation
		10.13.6 Vibrating String with Zero Initial Velocity
		10.13.7 Vibrating String with Given Initial Velocity and Zero Initial Displacement
		10.13.8 Vibrating String with Initial Displacement and Initial Velocity
		10.13.9 Laplace’s Equation or Potential Equation or Two-dimensional Steady-state Heat Flow Equation
	Exercise 10.7
	Exercise 10.8
	10.14 Two-dimensional Wave Equation
	Exercise 10.9
Chapter 11: Z-Transforms and Solution of Difference Equations
	11.1 Introduction
	11.2 Z-Transform: Definition
		11.2.1 Two Special Sequences
		11.2.2 Z-Transforms of Unit Step and Unit Impulse Sequences
	11.3 Z-Transforms of Some Standard Functions (Special Sequences)
		11.3.1. Unit Constant Sequence
		11.3.2. Alternating Unit Constant Sequence
		11.3.3. Geometric Sequence
		11.3.4. Natural Number Sequence
		11.3.5. Reciprocal Factorial Sequence
		11.3.6. Power-Cum-Reciprocal Factorial Sequence
		11.3.7. Binomial Coefficient Sequence
		11.3.8. Power-Cum-Reciprocal Factorial Sequence with a Multiple
	11.4 Recurrence Formula for the Sequence of a Power of Natural Numbers
	11.5 Properties of Z-Transforms
		11.5.1. Linearity
		11.5.2. Change of Scale or Damping Rule
		11.5.3. Shifting Property
		11.5.4. Multiplication by n
		11.5.5. Division by n
		11.5.6. Initial Value Theorem
		11.5.7. Final Value Theorem
		11.5.8. Convolution Theorem
	Exercise 11.1
	11.6 Inverse Z-Transform
		11.6.1 Methods for Evaluation of Inverse Z-Transforms
	Exercise 11.2
	11.7 Application of Z-Transforms: Solution of a Difference Equations by Z-Transform
		11.7.1 Introduction
		11.7.2 Difference Equation
		11.7.3 Order of a difference equation
		11.7.4 General Solution (Complete Solution)
		11.7.5 Particular Solution (Particular Integral)
		11.7.6 Linear Difference Equation
		11.7.7 Complementary Function and Particular Integral
	11.8 Method for Solving a Linear Difference Equation with Constant Coefficients
		11.8.1 (A) Complementary Function
		11.8.1 (B) Particular Integral
		11.8.2 Short Methods for Finding the Particular Integral
	Exercise 11.3
Chapter 12: Special Functions
	12.1 Introduction
	12.2 Gamma Function
	12.3 Recurrence Relation or Reduction Formula
		12.3.1 Gamma Function for Negative Non-Integer Values
		12.3.2 Some Standard Results
	12.4 Various Integral Forms of Gamma Function
		12.4.1 Form I: Integral of Log Function
		12.4.2 Form II: Exponential Function
		12.4.3 Form III: Scaling of Variable of Integration
		12.4.4 Form IV: The Product of a Power Function and a Logarithmic Function
		12.4.5 Form V: Product of a Power Function and an Exponential Function
	Exercise 12.1
	12.5 Beta Function
		12.5.1 Some Standard Results
	12.6 Various Integral Forms of Beta Function
		12.6.1 Form I: Beta Function as an Infinite Integral
		12.6.2 Form II: Beta Function in Symmetric Integral Form
		12.6.3 Form III: Improper Integral Form
		12.6.4 Form IV: Integral from 0 to 1 Form
		12.6.5 Form V: Integral from a to b Form
		12.6.6 Form VI: Integral of Circular Functions
		12.6.7 Form VII: Relation of Proportionality
		12.6.8 Form VIII: Beta Function in Explicit Form
	12.7 Relation Between Beta and Gamma Functions
	12.8 Multiplication Formula
	12.9 Legendre’s Duplication Formula
		12.9.1 Dirichlet’s Integral
	Exercise 12.2
	12.10 Legendre Functions
		12.10.1 Introduction
		12.10.2 Power Series Method of Solution of Linear Differential Equations
		12.10.3 Existence of Series Solutions: Method of Frobenius
		12.10.4 Legendre Functions
		12.10.5 Legendre Polynomials Pn(x)
		12.10.6 Generating Function for Legendre Polynomials Pn(x)
		12.10.7 Recurrence Relations of Legendre Functions
		12.10.8 Orthogonality of Functions
		12.10.9 Orthogonality of Legendre Polynomials Pn(x)
		12.10.10 Betrami’s Result
		12.10.11 Christoffel’s Expansion
		12.10.12 Christoffel’s Summation Formula
		12.10.13 Laplace’s First Integral for Pn(x)
		12.10.14 Laplace’s Second Integral for Pn(x)
		12.10.15 Expansion of f x) in a Series of Legendre Polynomials
	Exercise 12.3
	12.11 Bessel Functions
		12.11.1 Introduction
		12.11.2 Bessel Functions
		12.11.3 Bessel Functions of Non-Integral Order p : J p(x) and J − p(x)
		12.11.4 Bessel Functions of Order Zero and One: J0(x), J1 (x)
		12.11.5 Bessel Function of Second Kind of Order Zero Y0(x)
		12.11.6 Bessel Functions of Integral Order: Linear Dependence of Jn(x) and J −n(x)
		12.11.7 Bessel Functions of the Second Kind of Order n: Yn(x): Determination of Second Solution Yn(x) by the Method of Variation of Parameters
		12.11.8 Generating Functions for Bessel Functions
		12.11.9 Recurrence Relations of Bessel Functions
		12.11.10 Bessel’s Functions of Half- Integral Order
		12.11.11 Differential Equation Reducible to Bessel’s Equation
		12.11.12 Orthogonality Definition of orthogonality of functions
		12.11.13 Integrals of Bessel Functions
		12.11.14 Expansion of Sine and Cosine in Terms of Bessel Functions
	Exercise 12.4
	Exercise 12.5
Chapter 13: Functions of a Complex Variable
	13.1 Introduction
	13.2 Complex Numbers–Complex Plane
		13.2.1 Complex Function
		13.2.2 Limit of a Function
		13.2.3 Continuity at z0
		13.2.4 Differentiability
		13.2.5 Analytic Functions: Definition of Analyticity
		13.2.6 Cauchy–Riemann Equations
		13.2.7 Cauchy–Riemann Equations in Cartesian Coordinates
		13.2.8 Cauchy–Riemann Equations in Polar Coordinates
		13.2.9 Orthogonal Trajectories
	Exercise 13.1
	Exercise 13.2
	13.3 Laplace’s Equation: Harmonic and Conjugate Harmonic Functions
		13.3.1 Harmonic and Conjugate Harmonic Functions
	Exercise 13.3
Chapter 14: Elementary Functions
	14.1 Introduction
	14.2 Elementary Functions of a Complex Variable
		14.2.1 Exponential Function
		14.2.2 Trigonometric Functions
		14.2.3 Hyperbolic Functions
		14.2.4 Logarithm
		14.2.5 General powers of z: zα (α ∈ C)
		14.2.6 Inverse Trigonometric (Circular) Functions and Inverse Hyperbolic Functions
	Exercise 14.1
Chapter 15: Complex Integration
	15.1 Introduction
	15.2 Basic Concepts
		15.2.1 Simple Curve
		15.2.2 Closed Curve
		15.2.3 Smooth Curve or Arc
		15.2.4 Contour
		15.2.5 Simply-Connected Domain
		15.2.6 Multiply-Connected Domain
	15.3 Complex Line Integral
		15.3.1 Definition of the Complex Line Integral
		15.3.2 Properties
		15.3.3 Relation Between Real and Complex Line Integrals
		15.3.4 Evaluation of Complex Line Integral
		15.3.5 Analytic Functions: Path Independence
		15.3.6 Non-Analytic Functions: Path Dependence
	15.4 Cauchy–Goursat Theorem
		15.4.1 Alternative Statement of Cauchy’s Theorem
	15.5 Cauchy’s Theorem for Multiply-Connected Domain
		15.5.1 Basic Result: Integral of z = 1 Around
		15.5.2 Integral of Integer Power of (z-a) Around Circle of Radius r
		15.5.3 Evaluation of Complex Line Integral—Method 1
		15.5.4 Value of Line Integral: Independence of Path
		15.5.5 Integral of Non-Analytic Function: Dependence on Path of Integration
		15.5.6 Bound for the Absolute Value of an Integral (ML-Inequality)
		15.5.7 Verification and Application of C.I.T.
		15.5.8 Non-Analytic Functions
		15.5.9 Principle of deformation of path:
	15.6 Cauchy’s Integral Formula (C.I.F.) or Cauchy’s Formula Theorem
		15.6.1 Derivatives of Analytic Function (Cauchy’s Generalised Integral Formula)
		15.6.2 Statement of Cauchy’s Generalised Integral Formula
	15.7 Morera’s Theorem (Converse of Cauchy’s Theorem)
	15.8 Cauchy’s Inequality
	Exercise 15.1
Chapter 16: Complex Power Series
	16.1 Introduction
	16.2 Sequences and Series
	16.3 Power Series
	16.4 Series of Complex Functions
	16.5 Uniform Convergence of a Series of Functions
	16.6 Weierstrass’s M-Test
	16.7 Taylor’s Theorem (Taylor Series)
		16.7.1 Important Special Taylor Series
	16.8 Laurent’s Series
	16.9 Higher Derivatives of Analytic Functions
	Exercise 16.1
Chapter 17: Calculus of Residues
	17.1 Evaluation of Real Integrals
		17.1.1 Introduction
		17.1.2 Zeros and Singularities
		17.1.3 Types of Singularities
		17.1.4 Formulas for Residues at Poles
		17.1.5 Cauchy’s Residue Theorem
		17.1.6 Type I: Integrals of the Type
		17.1.7 Type II: Integral of the Type
		17.1.8 Type II (a): Improper Integrals Involving Trigonometric Functions
		17.1.9 Jordan’s Lemma
		17.1.10 Type III: Application of Jordan’s Lemma
		17.1.11 Type IV: Poles on the Real Axis (Indentation)
	Exercise 17.1
	Exercise 17.2
	Exercise 17.3
Chapter 18: Argument Principle and Rouche’s Theorem
	18.1 Introduction
	18.2 Meromorphic Function
	18.3 Argument Principle (Repeated Single Pole/Zero)
	18.4 Generalised Argument Theorem
	18.5 Rouche’s Theorem
	18.6 Liouville Theorem
	18.7 Fundamental Theorem of Algebra
	18.8 Maximum Modulus Theorem for Analytic Functions
	Exercise 18.1
Chapter 19: Conformal Mapping
	19.1 Introduction
		19.1.1 Mapping f: z ã w
		19.1.2 Conformal Mapping
	19.2 Conformal Mapping: Conditions for Conformality
	19.3 Conformal Mapping by Elementary Functions
		19.3.1 General Linear Transformation
		19.3.2 Inversion Transformation
	19.4 Some Special Transformations
		19.4.1 Transformation w = z2
		19.4.2 Transformation w = zn
		19.4.3 Transformation w = ez
		19.4.4 Transformation w = sin z
		19.4.5 Transformation w = cos z
		19.4.6 Transformation w = sinh z
		19.4.7 Transformation w = cosh z
		19.4.8 Logarithm
		19.4.9 Transformation w = z + 1/z (Joukowski1 Airfoil)
	19.5 Bilinear or Mobius or Linear Fractional Transformations
	19.6 Fixed Points of the Transformation
	Exercise 19.1
Question Bank
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Index