Contents

• 1. Fourier Series

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  Introduction, Fourier Series, Dirichlet’s Conditions, Even and Odd Functions, Fourier Series for Even and Odd Functions, Illustrative Examples, Exercise 1 (A), Fourier Series in Arbitrary Interval (0, 2l) or (–l, l), Half-Range Series, Illustrative Examples, Exercise 1 (B), Harmonic Analysis, Illustrative Examples, Exercise 1 (C).
   
  2. Z-Transform

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  Introduction, Definition, Properties of Z-Transforms, Linearity, Change of Scale or Damping Rule, Shifting Property, Multiplication by n, Division by n, Initial Value Theorem, Final Value Theorem, Illustrative Examples, Exercise 2 (A), Region of Convergence (R.O.C.), Inverse Z-Transform, Partial Fraction Method, Convolution Property for Z-Transform, Convolution Theorem for Casual Sequence, Solution of Difference Equations, Illustrative Examples, Exercise 2 (B).
   
  3. Laplace Transform

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  Introduction, Laplace Transform, Piecewise Continuous Function, Functions of Exponential Order, Sufficient conditions for the Existence of Laplace Transform, Laplace Transform of Some Elementary Functions, Properties of Laplace Transform, Linearity Property, First Shifting (or First Translation) Theorem, Second Shifting (or Second Translation) Theorem, Change of Scale Property, Laplace Transform of Derivatives, Laplace Transform of Integrals, Laplace Transform of f(t) Multiplied by Some Positive Integral Power of t, Laplace Transform of f(t)/t, Unit Step Function, Properties of Unit Step Function, Dirac Delta Function or Unit Impulse Function, Laplace Transform of Periodic Functions, Laplace Transform of Bessel Function, Illustrative Examples, Exercise 3 (A), Inverse Laplace Transform, Table of Inverse Laplace Transform, Properties of Inverse Laplace Transform, Linearity Property, First Shifting (or First Translation) Theorem, Second Shifting (or Second Translation) Theorem, Change of Scale Property, Inverse Laplace Transform of Derivatives, Inverse Laplace Transform of Integrals, Inverse Laplace Transform of F(s) Multiplied by Powers of s, Inverse Laplace Transform of f s s ( ), Convolution Theorem For Laplace Transform, Partial Fraction Method to find Inverse Laplace Transform, Illustrative Examples, Exercise 3 (B), Application of Laplace Transform to Ordinary Differential Equations with Constant Coefficients, Application of Laplace Transform to the System of Simultaneous Differential Equations, Application of Laplace Transform to Electrical Circuits, Illustrative Examples, Exercise 3 (C), Application of Laplace Transform to Partial Differential Equations, The Complex Inversion Formula, Illustrative Examples, Exercise 3 (D).
   
  4. Fourier Transform

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  Introduction, Fourier Integral, Fourier Integral Theorem, Equivalent Complex form or Exponential Form of Fourier Integral Theorem, Particular Cases of Fourier Integral Theorem, Fourier Transform Pairs, Fourier Transform or Complex Fourier Transform, Fourier Cosine Transform (FCT), Fourier Sine Transform (FST), Relationship Between Fourier and Laplace Transforms, Illustrative Examples, Exercise 4 (A), Properties of Fourier Transform, Convolution, Convolution Theorem for Fourier Transforms, Parseval’s Identity for Fourier Transforms, Applications of Fourier Transforms to Boundary Value Problems, Illustrative Examples, Exercise 4 (B).
   
  5. Finite Differences and Interpolation

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  Introduction, Finite Difference Calculus, Forward Differences, Backward Differences, Central Differences, Other Difference Operators, Relation Between Difference Operators, Exercise 5 (A), Errors in Polynomial Interpolation, Newton Gregory Forward Interpolation Formula, Newton-Gregory Backward Interpolation Formula, Central Difference Interpolation Formula, Lagrange’s Interpolation Formula for Unequal Spaced Points, Numerical Differentiation, Maximum and Minimum Value of a Tabulated Function, Numerical Integration, Illustrative Examples, Exercise 5 (B).
   
  6. Numerical Solution to Ordinary Differential Equations

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  Introduction, Picard’s Method, Euler’s Method, Modified Euler’s Formula, Runge-Kutta Method of Fourth Order, Milne’s Predictor Corrector Method, Illustrative Examples, Exercise 6 (A).
   
  P. Papers