At the end of this course, the students will be able to 1. Use the concepts and methods of complex numbers, complex functions, multi-valuedness, branch points and branch cuts, complex differential calculus, analytic functions, complex integral calculus, Laurent series and series expansions, residue theory and applications to real integrals, Laplace and Fourier transform inversions, Conformal mapping. 2. Use the concepts and methods of variational calculus, extremalization of functionals, essential and natural boundary conditions, extremalization of functionals with constraints, method of Lagrange multipliers, extremalization of functionals with variable boundaries, weak form of differential equations, Rayleigh-Ritz method. 3. Use the concepts and methods of integral equations, classifications, Fredholm and Volterra type integral equations, integral equations with separable (degenerate) kernel, solution techniques, Fredholm alternative.