Review of the basics of complex numbers. Definition of the elementary transcendent functions of a complex variable. Holomorphic functions: Cauchy-Riemann conditions, and conformal representations. Rules of derivation. Inverse functions, roots, and logarithms. The extended complex plane, and the Riemann sphere. Notion of path in a metric space. Regular walks in the complex plane. The path integral for functions of n real variables, and for functions of one complex variable. Conservative fields. Holomorphic functions, such as vector fields. Cauchy's theorem; Integrals of the Cauchy type, and the Cauchy integral formula. Harmonic functions. Principle of the highest modulus. Liouville's theorem. Power series; analytic functions. Notable series. Isolated singularities; Laurent expansion, classification of isolated singularities. Theorem of residues, and its applications. Fundamental Theorem on Analytical Prolongation; analytic extension along a path; monodromy, and polydromy; complete analytic functions, and Riemann surface. Integrals of polydrome functions. The Gamma function. Ordinary, linear, 2nd order differential equations. Local existence and uniqueness theorem; analytical extension of the solutions; structure of the solution space. Singular points. Euler's equation. Fundamental solutions. Regular singularities, and behaviour of the solutions in their neighbourhood. Bessel equation and functions. Equations of the Fuchs class; Gauss equation; Hypergeometric function; confluent hypergeometric equation.