## 21MAT41

## Module - 1

Complex Analysis: Review of a function of a complex variable, limits, continuity and differentiability. Analytic functions: Cauchy-Riemann equations in cartesian and polar forms and consequences. Construction of analytic functions by Milne-Thomson method, Problems.

Complex integration: Line integral of a complex function, Cauchy’s theorem and Cauchy’s integral formulaÂ andÂ problems.

## Module - 2

Special functions: Series solution of Bessel’s differential equation leading to In(x) Bessel’s function of the first kind, Properties, Orthogonality of Bessel’s functions. Series solution of Legendre’s differential equation leading to Pn (x)-Legendre polynomials. Rodrigue’s formula (without proof),Â problems.

## Module - 3

Statistical Methods: Correlation and regression-Karl Pearson’s coefficient of correlation and rank correlation, problems. Regression analysis, lines of regression, problems.

Curve Fitting: Curve fitting by the method of least squares, fitting the curves of the forms

y = ax + b, y = axb and yÂ =Â ax2Â +Â bxÂ +Â c.

## Module - 4

Probability Distributions: Review of basic probability theory. Random variables (discrete and continuous), probability mass and density functions. Mathematical expectation, mean and variance. Binomial, Poisson and normal distributions- problems (derivations for mean and standard deviation for Binomial and Poisson distributions only)-IllustrativeÂ examples.

## Module - 5

Joint probability distribution: Joint Probability distribution for two discrete random variables, expectation, covariance and correlation.

Sampling Theory: Introduction to sampling distributions, standard error, Type-I and Type-Il errors. Test of hypothesis for means, student’s t-distribution, Chi-square distribution as a test of goodnessÂ ofÂ fit.