Course Contents :
Complex Variables Analytic functions of a complex variable. Cauchy-Riemann conditions. Power series. Cauchy?s integral theorem. Conformal mapping. Singularities: poles, essential singularities. Residue theorem. Contour integration and examples. Analytic continuation. Multiple-valued functions, branch points and branch cut integration. Partial Differential Equations : Partial differential equations in Physics: Laplace, Poisson and Helmholtz equations; diffusion and wave equations. Applications. Integral transforms : Laplace transforms and Fourier transforms. Parsevall?s theorem. Convolution theorem. Applications. Calculus of Variations Functionals. Natural boundary conditions. Lagrange multipliers. Rayleigh-Ritz method. Group theory : Elements of group theory. Discrete groups with examples. Contionuos groups (Lie groups) [rotation group in 2 and 3 dimensions, U(1) and SU(2)]. Generators. Representations, Character tables for some point groups and the orthogonality theorem.
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