Analytic functions. Cauchy-Riemann equations. Harmonic functions. Elementary functions: the exponential function, trigonometric functions, hyperbolic functions. The logarithmic function and its branches. Contour Integrals and Cauchy’s theorem. Cauchy integral formula. Liouville’s theorem and the fundamental theorem of algebra. Maximum moduli of functions. Incompressible and irrotational flow. Complex potential. Laurent’s series and classification of singularities. Sources and vortices as singular points of potential flow. Calculus of residues. Conformal mappings. Fractional linear transformations. Applications of conformal mappings. Laplace’s equation. Electrostatic potential. Elements of elliptic functions. Analytic continuation and elementary Riemann surfaces.