Complex analysis can be seen as an application of mathematical analysis to complex functions f(z) with variable z. The results obtained in the complex field have a lot of applications and are able to provide a deeper understanding of mathematics, also with regard to real functions, solving their intrinsic problems, such as the calculation of the square root of a negative number. The theory of complex analysis is more subtle than that dedicated to functions with two real variables, although a complex number has two degrees of freedom: its real part x and its imaginary part y, in the Cartesian representation z=x+iy. For example, the differentiability of a complex function requires additional conditions with respect to the simple existence and continuity of the partial derivatives with respect to the real and imaginary parts. In this theoretical text we expose the fundamental results representing the foundations of all the complex analysis. The general topics are: complex numbers and their properties, functions in the complex field, analytic functions and their singularities, integration and series in the complex field, Taylor series and Laurent series, residues and their calculation techniques, multivalued functions, dispersion relations, Euler's gamma function. Among the most important theorems and results we find: Cauchy's theorem, Cauchy's integral representation, Morera's theorem, residue theorem, Jordan's lemma, Schwarz's reflection principle, Mittag-Leffler expansion.
