Integrals of complex functions of a real variable. Paths, closed paths, simple closed paths. Piecewise regular paths. Trace of a path. Equivalent paths: curves. Length of a piecewise regular path and its invariance under reparametrization. Jordan curve theorem (not proved). Integrals of complex functions along piecewise regular paths. Invariance of the integral under reparametrization of the path. Upper bound of the modulus of an integral: Darboux inequality. Theorem on the existence of a primitive. Primitives of multi-valued functions and calculation of integrals on closed curves. Cauchy-Goursat theorem for simple closed curves. Homotopic curves. Closed curves homotopic to zero. Simply connected open sets. Extension of the Cauchy-Goursat theorem to closed curves in simply connected domains. An analytic function in a simply connected domain admits primitive. Principle of deformation of paths. Cauchy’s integral formula for an analytic function. Existence and Cauchy’s integral formula for the derivatives of any order of analytic functions. The derivative of any order of an analytic function is analytic. Morera’s theorem. Cauchy’s inequalities. Liouville’s theorem. Fundamental theorem of algebra: decomposition of a polynomial.