Numerical Methods: Numerical solution of ordinary differential equations of first order and first degree, Taylor’s series method, modified Euler’s method, Runge - Kutta method of fourth order.Milne’s and Adams-Bashforth predictor and corrector methods (No derivations of formulae).
Numerical Methods: Numerical solution of second order ordinary differential equations, Runge-Kutta method and Milne’s method.Special Functions: Series solution-Frobenious method. Series solution of Bessel’s differential equation leading to Jn(x)-Bessel’s function of first kind. Basic properties and orthogonality. Series solution of Legendre’s differential equation leading to Pn(x)-Legendre polynomials. Rodrigue’sformula, problems
Complex Variables: Review of a function of a complex variable, limits, continuity, differentiability. Analytic functions-Cauchy-Riemann equations in cartesian and polar forms. Properties and construction of analytic functions. Complex line integrals-Cauchy’s theorem and Cauchy’s integral formula, Residue, poles, Cauchy’s Residue theorem ( without proof) and problems.Transformations: Conformal transformations, discussion of transformations: w=z2, w=e2, w=z+(1/z)(z≠0) and bilinear transformations-problems
Probability Distributions: Random variables (discrete and continuous), probability mass/density functions. Binomial distribution, Poisson distribution. Exponential and normal distributions, problems.Joint probability distribution: Joint Probability distribution for two discrete random variables, expectation, covariance, correlation coefficient.
Sampling Theory: Sampling, Sampling distributions, standard error, test of hypothesis for means and proportions, confidence limits for means, student’s t-distribution, Chi-square distribution as a test of goodness of fit.Stochastic process:Stochastic processes, probability vector, stochastic matrices, fixed points, regular stochastic matrices, Markov chains, higher transition probability simple problems.