L 3 C 3
Teachers Continuous Evaluation: 40 marks. Term-End Semester Examination: 60 marks.
Course outcomes
- Apply elementary probability and statistical methods to analyze engineering data, quantify uncertainty, and support engineering decision-making.
- Apply the principles of complex analysis to solve engineering problems involving analytic functions and contour integration.
- Employ Laplace and Fourier analytical techniques for solving ordinary differential equations and engineering models.
- Formulate and solve classical partial differential equations arising in heat transfer, wave propagation, and related engineering applications.
Unit I
Probability concepts, Conditional probability, Bayes' theorem, Random variables, Discrete and continuous probability distributions, Binomial, Poisson and Normal distributions, Mathematical expectation and variance, Correlation and regression, Sampling distributions, Confidence intervals, Tests of hypotheses for means and proportions, Engineering applications.
Unit II
Complex functions, Analytic functions, Cauchy–Riemann equations, Harmonic functions and Laplace equation, Exponential, trigonometric, hyperbolic and logarithmic functions, Euler's formula, De Moivre's theorem (without proof), General power and principal value, Singularities and zeros, Line integrals in the complex plane, Cauchy's integral theorem, Cauchy's integral formula, Taylor and Laurent series, Residue theorem, Evaluation of definite real integrals using residues.
Unit III
Laplace transforms: Definition, properties, transforms of derivatives and integrals, Shifting theorems, Unit step function, Dirac delta function, Inverse Laplace transforms, Convolution theorem, Solution of ordinary differential equations. Fourier analysis: Fourier series, Functions of arbitrary period, Half-range expansions, Sturm–Liouville problems, Introduction to Fourier transforms and engineering applications.