Engineering Mathematics: Volume - III

Description

Cover Engineering Mathematics-III Copyright Contents About the Authors Preface 1. Special Functions 1.1 Introduction 1.2 Gamma Function 1.3 Recurrence Relation or Reduction Formula 1.4 Various Integral Forms of Gamma Function Exercise 1.1 1.5 Beta Function 1.6 Various Integral Forms of Beta Function 1.7 Relation Between Beta and Gamma Functions 1.8 Multiplication Formula 1.9 Legendre’s Duplication Formula Exercise 1.2 1.10 Legendre Functions Exercise 1.3 1.11 Bessel Functions Exercise 1.4 2. Functions of a Complex Variable 2.1 Introduction 2.2 Complex Numbers— Complex Plane Exercise 2.1 Exercise 2.2 2.3 Laplace’s Equation: Harmonic and Conjugate Harmonic Functions Exercise 2.3 3. Elementary Functions 3.1 Introduction 3.2 Elementary Functions of a Complex Variable Exercise 3.1 4. Complex Integration 4.1 Introduction 4.2 Basic Concepts 4.3 Complex Line Integral 4.4 Cauchy–Goursat Theorem 4.5 Cauchy’s Theorem for Multiply-Connected Domain Theorem 4.6 Cauchy’s Integral Formula (C.I.F.) or Cauchy’s Formula Theorem 4.7 Morera’s Theorem (Converse of Cauchy’s Theorem) 4.8 Cauchy’s Inequality Exercise 4.1 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) 5.8 Laurent Series 5.9 Higher Derivatives of Analytic Functions Exercise 5.1 6. Calculus of Residues 6.1 Evaluation of Real Integrals Exercise 6.1 Exercise 6.2 Exercise 6.3 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 8. Conformal Mapping 8.1 Introduction 8.2 Conformal Mapping: Conditions for Conformality 8.3 Conformal Mapping by Elementary Functions 8.4 Some Special Transformations 8.5 Bilinear or Mobius or Linear Fractional Transformations 8.6 Fixed Points of the Transformation w = (az+b)/(cz+d) Exercise 8.1 Question Bank Multiple Choice Questions Fill in the Blanks Match the Following True or False Statements Question Papers Bibliography Index