This unit provides the basic tools in the use of complex variables to solve practical problems in Applied Mathematics and Physics. It continues the use of integral transform methods to solve ordinary and partial differential equations. Analytic functions. The elementary functions. Contour integrals, Cauchy's integral theorem and the integral formula. Taylor and Laurent series. The residue theorem. Evaluating real integrals using complex methods. Conformal mapping and applications. Differential equations with regular singular points and the Frobenius method. Special functions. The Fourier transform and inverse. Evaluating Fourier transform using complex variable theory. Convolution theorem. Solving PDE's using Fourier transforms. Fourier transform of generalized functions. The Laplace transform and its inversion formula (the Bromwich integral). The convolution theorem. Solving ODE's and PDE's using Laplace transforms. Integral Equations. Fredholm and Volterra integral equations. Fredholm integral equations with degenerate kernels. Integral equations with convolution kernels. Asymptotic Approximation Methods. Method of steepest descent. Method of stationary phase.