Complex Analysis: introduction, Cartesian and Polar representation, Euler formula, differentiability, Cauchy-Riemann equation, contour integration, Cauchy-Goursat theorem, Cauchy integral formula, Taylor and Laurent series, Cauchy residue theorem. Fourier Series: definition, Fourier exponential series, trigonometric Fourier series; Fourier Transform: Integral transform, Fourier integral, inverse Fourier transform, properties of Fourier transform, Perseval’s theorem; Poisson’s summation formula, Fourier transform in Rd. Vector Analysis: derivatives, and integrals of vector valued functions, gradient vectors and tangent planes; Multivariable Calculus: partial differentiation, functions of several variables, limits and continuity, partial derivatives, vector fields, gradient, divergence, and curl, line, double, and multiple integrals, change of variable in multiple integrals, Jacobian, Green’s theorem, Stokes theorem, Gauss’s theorem. Beta and Gamma Function: beta and gamma integrals, Euler product, reflection formula, duplication formula, asymptotics, digamma function, Hurwitz and Riemann zeta function; Series Solution of Second Order ODEs: Frobenius’s method, series around a regular point, expansion around a regular singular point; Bessel Function: differential equation, generating function, Hankel function, asymptotic expansion; Legendre Polynomial: differential equation, Legendre function of first and second kind, generating function, associated Legendre polynomial, spherical harmonics.