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# 21MAT41 – M4 VTU Notes | 2021 Scheme – 4th Sem

VTU NOTES
Subject code
21MAT41
Semester
4th

## 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 formula and 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 Pr (x)-Legendre polynomials. Rodrigue’s formula (without proof), problems.

## Module - 3

Statistical Methods: Correlation and regression- Karl Person’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

## 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, Poison and normal distributions- problems (derivations for mean and standard deviation for Binomial and Poison distributions only]-Illustrative 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, and Chi-square distribution as a test of goodness of fit.