Engineering Mathematics Volume III

Description

Cover About the Author Contents Preface Chapter 1: Special Functions 1.1 Introduction 1.2 Gamma Function 1.3 Recurrence Relation or Reduction Formula 1.3.1 Gamma Function for Negative Non-Integer Values 1.3.2 Some Standard Results 1.4 Various Integral Forms of Gamma Function 1.4.1 Form I: Integral of Log Function 1.4.2 Form II: Exponential Function 1.4.3 Form III: Scaling of Variable of Integration 1.4.4 Form IV: The Product of a Power Function and a Logarithmic Function 1.4.5 Form V: Product of a Power Function and an Exponential Function Exercise 1.1 1.6 Various Integral Forms of Beta Function 1.6.1 Form I: Beta Function as an Infinite Integral 1.6.2 Form II: Beta Function in Symmetric Integral Form 1.6.3 Form III: Improper Integral Form 1.6.4 Form IV: Integral from 0 to 1 Form 1.6.5 Form V: Integral from a to b Form 1.6.6 Form VI: Integral of Circular Functions 1.6.7 Form VII: Relation of proportionality 1.6.8 Form VIII: Beta Function in Explicit Form 1.7 Relation Between Beta and Gamma Functions 1.8 Multiplication Formula 1.9 Legendre’s Duplication Formula 1.9.1 Dirichlet’s Integral Exercise 1.2 1.10 Legendre Functions 1.10.1 Introduction 1.10.2 Power Series Method of Solution of Linear Differential Equations 1.10.3 Existence of Series Solutions: Method of Frobenius 1.10.4 Legendre Functions 1.10.5 Legendre Polynomials Pn(x) 1.10.6 Generating Function for Legendre Polynomials Pn(x) 1.10.7 Recurrence Relations of Legendre Functions 1.10.8 Orthogonality of Functions 1.10.9 Orthogonality of Legendre Polynomials Pn(x) 1.10.10 Betrami’s Result 1.10.11 Christoffel’s Expansion 1.10.12 Christoffel’s Summation Formula 1.10.13 Laplace’s First Integral for Pn(x) 1.10.14 Laplace’s Second Integral for Pn(x) 1.10.15 Expansion of f (x) ina Series of Legendre Polynomials Exercise 1.3 1.11 Bessel Functions 1.11.1 Introduction 1.11.2 Bessel Functions 1.11.3 Bessel Functions of Non-Integral Order p :Jp(x) and J−p(x) 1.11.4 Bessel Functions of Order Zero and One: J0(x), J1(x) 1.11.5 Bessel Function of Second Kind of Order Zero Y0(x) 1.11.6 Bessel Functions of Integral Order: Linear Dependence of Jn(x) and J−n(x) 1.11.7 Bessel Functions of the Second Kind of Ordern : Yn(x): Determination of Second Solution Yn(x) by the Method of Variation of Parameters 1.11.8 Generating Functions for Bessel Functions 1.11.9 Recurrence Relations of Bessel Functions 1.11.10 Bessel’s Functions of Half- Integral Order 1.11.11 Differential Equation Reducible to Bessel’s Equation 1.11.12 Orthogonality 1.11.13 Integrals of Bessel Functions 1.11.14 Expansion of Sine and Cosine in Terms of Bessel Functions Exercise 1.4 Chapter 2: Functions of a Complex Variable 2.1 Introduction 2.2 Complex Numbers–Complex Plane 2.2.1 Complex Function 2.2.2 Limit of a Function 2.2.3 Continuity at z0 2.2.4 Differentiability 2.2.5 Analytic Functions: Definition of Analyticity 2.2.6 1Cauchy–2Riemann Equations 2.2.7 Cauchy–Riemann Equations in Cartesian Coordinates 2.2.8 Cauchy–Riemann Equations in Polar Coordinates 2.2.9 Milne–Thomson’s Method 2.2.10 Orthogonal Trajectories Exercise 2.1 Exercise 2.2 2.3 Laplace’s Equation: Harmonic and Conjugate Harmonic Functions 2.3.1 Harmonic and Conjugate Harmonic Functions Exercise 2.3 Chapter 3: Elementary Functions 3.1 Introduction 3.2 Elementary Functions of a Complex Variable 3.2.1 Exponential Function 3.2.2 Trigonometric Functions 3.2.3 Hyperbolic Functions 3.2.4 Logarithm 3.2.5 General powers of z:z (2 C) 3.2.6 Inverse Trigonometric (Circular) Functions and Inverse Hyperbolic Functions Exercise 3.1 Chapter 4: Complex Integration 4.1 Introduction 4.2 Basic Concepts 4.2.1 Simple Curve 4.2.2 Closed Curve 4.2.3 Smooth Curve or Arc 4.2.4 Contour 4.2.5 Simply-Connected Domain 4.2.6 Multiply-Connected Domain 4.3 Complex Line Integral 4.3.1 Definition of the Complex Line Integral 4.3.2 Properties 4.3.3 Relation Between Real and Complex Line Integrals 4.3.4 Evaluation of Complex Line Integral 4.3.5 Analytic Functions: Path Independence 4.3.6 Non-Analytic Functions: Path Dependence 4.4 Cauchy–Goursat Theorem 4.4.1 Alternative Statement of Cauchy’s Theorem 4.5 Cauchy’s Theorem for Multiply-Connected Domain Theorem 4.5.1 Basic Result: Integral of 1z Around |z| = 1 (Unit Circle) 4.5.2 Integral of Integer Power of (z − a) Around Circle of Radius r 4.5.3 Evaluation of Complex Line Integral—Method 1 4.5.4 Value of Line Integral: Independence of Path 4.5.5 Integral of Non-Analytic Function: Dependence on Path of Integration 4.5.6 Bound for the Absolute Value of an Integral (ML-Inequality) 4.5.7 Verification and Application of C.I.T. 4.5.8 Non-Analytic Functions 4.5.9 Principle of deformation of path: 4.6 Cauchy’s Integral Formula (C.I.F.) or Cauchy’s Formula Theorem 4.6.1 Derivatives of Analytic Function (Cauchy’s Generalised Integral Formula) 4.6.2 Statement of Cauchy’s Generalised Integral Formula 4.7 Morera’s Theorem (Converse of Cauchy’s Theorem) 4.8 Cauchy’s Inequality Exercise 4.1 Chapter 5: Complex Power Series 5.1 Introduction 5.2 Sequences and Series 5.3 Power Series 5.4 Series of Complex Functions 5.5 Uniform Convergence of a Series of Functions 5.6 Weierstrass’s M-Test 5.7 Taylor’s Theorem (Taylor Series) 5.7.1 Important Special Taylor Series 5.8 Laurent’s Series 5.9 Higher Derivatives of Analytic Functions Exercise 5.1 Chapter 6: Calculus of Residues 6.1 Evaluation of Real Integrals 6.1.1 Introduction 6.1.2 Zeros and Singularities 6.1.3 Types of Singularities 6.1.4 Formulas for Residues at Poles 6.1.5 Cauchy’s Residue Theorem 6.1.6 Type I: Integrals of the Type 6.1.7 Type II: Integral of the Type R 1−1f (x) dx 6.1.8 Type II (a): Improper Integrals Involving Trigonometric Functions 6.1.9 Jordan’s Lemma 6.1.10 Type III: Application of Jordan’s Lemma 6.1.11 Type IV: Poles on the Real Axis (Indentation) Exercise 6.1 Exercise 6.2 Exercise 6.3 Chapter 7: Argument Principle and Rouche’s Theorem 7.1 Introduction 7.2 Meromorphic Function 7.3 Argument Principle (Repeated Single Pole/Zero) 7.4 Generalised Argument Theorem 7.5 Rouche’s Theorem 7.6 Liouville Theorem 7.7 Fundamental Theorem of Algebra 7.8 Maximum Modulus Theorem for Analytic Functions Exercise 7.1 Chapter 8: Conformal Mapping 8.1 Introduction 8.1.1 Mapping f: z w 8.1.2 Conformal Mapping 8.2 Conformal Mapping: Conditions for Conformality 8.3 Conformal Mapping by Elementary Functions 8.3.1 General Linear Transformation 8.3.2 Inversion Transformation 8.4 Some Special Transformations 8.4.1 Transformation w = z2 8.4.2 Transformation w = zn (n 2 N) 8.4.3 Transformation w = ez 8.4.4 Transformation w = sin z 8.4.5 Transformation w = cos z 8.4.6 Transformation w=sinh z 8.4.7 Transformation w=cosh z 8.4.8 Logarithm 8.4.9 Transformation w = z +1z (Joukowski1Airfoil 8.5 Bilinear or Mobius2 or Linear Fractional Transformations 8.6 Fixed Points of the Transformation Exercise 8.1 Question Bank Mulitple Choice Questions Fill in the blanks Match the Following True or False Statements Question Papers Bibliography Index