We provide a revision of some basic facts about functions of a complex variable. This revision includes the main properties of such functions in what concerns their differentiation and integration and which will be useful, for instance, when we study integral transforms, in which the obtention of the final results will often require the calculation of integrals using the method of residues. We present Cauchy–Riemann conditions, Laurent series, zeros, and singularities, where the branch point appears as a possibility. The residue theorem and the Jordan lemma play an important rule in the evaluation of some real integrals. Some solved exercises are discussed step by step, specifically involving the Bromwich contour and the modified Bromwich contour. As an application, we discuss some real integrals where the gamma function appears as a natural way. We conclude with a list of proposed exercises; among them, some interesting applications are left to the reader, particularly the ones involving different contours to evaluate real integrals by means of complex functions, all of them with the answer and/or suggestion.