Engineering Mathematics: Volume - III

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