1. Fourier Transform
Introduction, Fourier Integral, Fourier Integral Theorem, Equivalents 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 1 (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 1 (B), Discrete Fourier Transform (DFT), Discrete Inverse Fourier Transform, Properties of Discrete Fourier Transform, Fast Fourier Transform Form Method, Fast Fourier Transform Algorithm, Fast Fourier algorithm for n when n is Product of Three Integers, Discrete Fourier Cosine Transform, Exercise 1 (C).
2. Laplace Transform
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 2 (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 2 (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 2 (C), Application of Laplace Transform to Partial Differential Equations, The Complex Inversion Formula, Illustrative Examples, Exercise 2 (D).
3. Random Variables and Discrete Probability Distributions
Introduction, Random Variable, Discrete Random Variable, Continuous Random Variable, Probability Distribution of a Discrete Random Variable, Probability Mass Function, Distribution Function, Probability Distribution of a Continuous Random Variable, Probability Density Function, Distribution Function, Expectation, Variance, Measures of Central Tendency, Measures of Dispersion, Moments, Moment About Origin, Moment About Mean or Central Moment, Moment About an Arbitrary Point, Karl Pearson ? and ? Coefficients, Moment Generating Function (mgf), Binomial Distribution, Mean and Variance of Binomial Distribution, Moments, Moment Generating Function and Recurrence Relation for Moments, Fitting of Binomial Distribution (Recurrence Relation for the Probabilities of Binomial Distribution), Poisson Distribution, Mean and Variance of Poisson Distribution, Moments, Moment Generating Function and Recurrence Relation for Moments, Recurrence Relation for Probabilities of Poisson Distribution or Fitting of Poisson Distribution, Geometric Distribution, Mean and Variance of Geometric Distribution, Moment Generating Function of Geometric Distribution, Illustrative Examples, Exercise 3 (A), Exercise 3 (B).
4. Continuous Probability Distributions & Correlation
Introduction, Normal Distribution, Standard Form of the Normal Distribution, Normal Probability Integral (Area Under the Standard Probability Curve), Mean and Variance of Normal Distribution, Moment Generating Function of Normal Distribution, Recurrence Relation for Even Order Central Moments, Fitting of Normal Distribution, Rectangular or Uniform Distribution, Moments and Moment Generating Function, Mean and Variance, Exponential Distribution, Moments, Moment Generating Function, Mean and Variance, Memory less Property of Exponential Distribution, Gamma Distribution, Illustrative Examples, Exercise 4 (A), Bivariate Distribution, Correlation, Measure of Correlation: Karl Pearson Coefficient of Correlation, Rank Correlation, Curve Fitting, Principle of Least Squares, Fitting of a Straight Line, Fitting a Parabola, Fitting of Other Curves, Regression, Linear Regression, Lines of Regression, Angle Between Two Lines of Regression, Illustrative Example, Exercise 4 (B).
5. Finite Differences and Interpolation
Introduction, Finite Difference Calculus, Forward Differences, Backward Differences, Central Differences, Other Difference Operators, Relation Between Difference Operators, Exercise 5 (A), Newton Gregory Forward Interpolation Formula, Newton-Gregory Backward Interpolation Formula, Stirling’s Central Difference Interpolation Formula, Lagrange’s Interpolation Formula for Unequal Interval, Illustrative Examples, Exercise 5 (B).
6. Numerical Solution to Non-Linear Equations in One Variable
Introduction, Algebraic and Transcendental Equations, Solution of Algebraic and Transcendental Equations, Exercise 6 (A).
7. Numerical Solution to Simultaneous Equations
Introduction, Gauss Elimination Method [Direct Method], Gauss Seidel Method [Iterative Method], Illustrative Examples, Exercise 7 (A).
8. Numerical Differentiation, Integration & Numerical Solution to O.D.E.
Introduction, Numerical Differentiation, Numerical Integration, Illustrative Examples, Exercise 8 (A), Solution to Ordinary Differential Equations, Picard’s Method, Euler’s Method, Modified Euler’s Method, Runge-Kutta Method of Fourth Order, Milne’s Predictor Corrector Method, Illustrative Examples, Exercise 8 (B).
A. Appendix
P. Paper