Section-A
Fourier Series and Fourier Transforms : Euler’s formulae, conditions for a Fourier
expansion, change of interval, Fourier expansion of odd and even functions, Fourier
expansion of square wave, rectangular wave, saw-toothed wave, half and full rectified
wave, half range sine and cosine series.
Fourier integrals, Fourier transforms, Shifting theorem (both on time and frequency
axes), Fourier transforms of derivatives, Fourier transforms of integrals, Convolution
theorem, Fourier transform of Dirac-delta function.
Section-B
Functions of Complex Variable: Definition, Exponential function, Trigonometric and
Hyperbolic functions, Logarithmic functions. Limit and Continuity of a function,
Differentiability and Analyticity.
Cauchy-Riemann equations, necessary and sufficient conditions for a function to be
analytic, polar form of the Cauchy-Riemann equations. Harmonic functions, application
to flow probles. Integration of complex functions. Cauchy-Integral theorem and formula.
Section-Cm
Power series, radius and circle of convergence, Taylor's Maclaurin's and Ls aurent’
series. Zeroes and singularities of complex functions, Residues. Evaluation of real
integrals using residues (around unit and semi circle only).
Probability Distributions and Hypothesis Testing: Conditional probability, Bayes
theorem and its applications, expected value of a random variable. Properties and
application of Binomial, Poisson and Normal distributions.
Section-D
Testing of a hypothesis, tests of significance for large samples, Student’ - s t
distribution (applications only), Chi-square test of goodness of fit.
Linear Programming: Linear programming problems formulation, solving linear
programming problems using (i) Graphical method (ii) Simplex method (iii) Dual
simplex method