Engineering Mathematics-III : ( Subject Code: 3EX1, 3EC1, 3EE6.1) For RTU

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Cover Contents Preface Roadmap to the Syllabus Symbols and Basic Formulae Chapter 1: Preliminaries 1.1 Sets and Functions 1.2 Continuous and Piecewise Continuous Functions 1.3 Derivability of a Function and Piecewise Smooth Functions 1.4 The Riemann Integral 1.5 The Causal and Null Function 1.6 Functions of Exponential Order 1.7 Periodic Functions 1.8 Even and Odd Functions 1.9 Sequence and Series 1.10 Series of Functions 1.11 Partial Fraction Expansion of a Rational Function 1.12 Special Functions 1.13 The Integral Transforms Chapter 2: Laplace Transform 2.1 Definition and Examples of Laplace Transform 2.2 Properties of Laplace Transforms 2.3 Limiting Theorems 2.4 Miscellaneous Examples Exercises Chapter 3: Inverse Laplace Transform 3.1 Definition and Examples of Inverse Laplace Transform 3.2 Properties of Inverse Laplace Transform 3.3 Partial Fractions Method to Find Inverse Laplace Transform 3.4 Heaviside’s Expansion Theorem 3.5 Series Method to Determine Inverse Laplace Transform 3.6 Convolution Theorem 3.7 Complex Inversion Formula 3.8 Miscellaneous Examples Exercises Chapter 4: Applications of Laplace Transform 4.1 Ordinary Differential Equations (a) Ordinary Differential Equations with Constant Solution (b) Problems Related to Electrical Circuits (c) Mechanical System (Mass-Spring System) (d) Ordinary Differential Equations with Polynomial (Variable) Coefficients 4.2 Simultaneous Differential Equations 4.3 Difference Equations 4.4 Integral Equations 4.5 Integro-Differential Equations 4.6 Solution of Partial Differential Equations 4.7 Evaluation of Integrals 4.8 Miscellaneous Examples Exercises Chapter 5: Fourier Transform 5.1 Fourier Integral Theorem 5.2 Fourier Transforms 5.3 Fourier Cosine and Sine Transforms 5.4 Properties of Fourier Transforms 5.5 Solved Examples 5.6 Complex Fourier Transforms 5.7 Convolution Theorem 5.8 Parseval’s Identities 5.9 Fourier Integral Representation of a Function 5.10 Finite Fourier Transforms 5.11 Applications of Fourier Transforms 5.12 Application to Differential Equations 5.13 Application to Partial Differential Equations Exercises Chapter 6: Fourier Series 6.1 Trigonometric Series 6.2 Fourier (or Euler) Formulae 6.3 Periodic Extension of a Function 6.4 Fourier Cosine and Sine Series 6.5 Complex Fourier Series 6.6 Spectrum of Periodic Functions 6.7 Properties of Fourier Coefficients 6.8 Dirichlet’s Kernel 6.9 Integral Expression for Partial Sums of a Fourier Series 6.10 Fundamental Theorem (Convergence Theorem) of Fourier Series 6.11 Applications of Fundamental Theorem of Fourier Series 6.12 Convolution Theorem for Fourier Series 6.13 Integration of Fourier Series 6.14 Differentiation of Fourier Series 6.15 Examples of Expansions of Functions in Fourier Series 6.16 Method to Find Harmonics of Fourier Series of a Function from Tabular Values 6.17 Signals and Systems 6.18 Classification of Signals 6.19 Classification of Systems 6.20 Response of a Stable Linear Time-Invariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input 6.21 Application to Differential Equations 6.22 Application to Partial Differential Equations 6.23 Miscellaneous Examples Exercises Chapter 7: Calculus of Variations 7.1 Functional 7.2 Closeness of Functions in the Sense of NTH Order Proximity 7.3 Extreme Values of the Functionals 7.4 Euler’s Equation 7.5 Stationary Functions in Particular Cases 7.6 Conditional Extremum 7.7 Euler–Poisson Equation for Functional Involving Higher-Order Derivatives Exercises Chapter 8: Functions of Complex Variables 8.1 Basic Concepts Logarithms of Complex Numbers Real and Imaginary Parts of Log (x + iy) Hyperbolic Functions Relations Between Hyperbolic and Circular Functions Periodicity of Hyperbolic Function 8.2 Analytic Functions 8.3 Integration of Complex-Valued Functions 8.4 Power Series Representation of an Analytic Function 8.5 Zeros and Poles 8.6 Residues and Cauchy’s Residue Theorem 8.7 Evaluation of Real Definite Integrals (A) Integration Around the Unit Circle (B) Definite Integral of the Type (C) Poles on the Real Axis 8.8 Conformal Mapping Bilinear (Mobius or Fractional) Transformation Particular cases of w 8.9 Miscellaneous Examples Exercises Chapter 9: The z-transform 9.1 Some Elementary Concepts 9.2 Definition of z-transform 9.3 Convergence of z-transform 9.4 Examples of z-transform 9.5 Properties of the z-transform 9.6 Inverse z-transform (A) Contour Integration Method (B) Partial Fractions Method (C) Power Series Method for Finding Inverse z-transform 9.7 Convolution Theorem 9.8 The Transfer Function (or System Function) 9.9 Systems Described by Difference Equations Exercises Solved Question Papers February 2010 Solutions February 2011 Solutions Index