Complex analysis - Holomorphic application theory

Complex numbers were first introduced in formulas for solving third- and fourth-degree equations using radicals (work by Cardan, Ferrari and Scipio del Ferro in the XVI ^e century). Until the advent of infinitesimal calculus, their existence was algebraic.

From this point of view alone, the fundamental theorem of algebra, which states that every polynomial admits a complex root, is already an issue that requires a thorough examination of the geometric nature of the complex field.

On the other hand, trigonometry led Euler to formulate his famous formulas linking the complex exponential to real circular lines. The precise study of functions of the complex variable dates from this period. It soon became clear that, while they posed no essentially new problems in terms of continuity, they required a different approach when it came to their derivability.

This is due to the very nature of the complex plane: unlike the real line, it cannot be provided with an ordered body structure, which is precisely what allows – 1 to find a square root, the number "i". On the other hand, a continuous application doesn't usually have a primitive in $ℂ$ : there, integration takes place along paths, and there are many paths leading from one point to another. In $ℝ$ , only one segment is suitable. These profound differences with regard to primitivation have consequences for the very notion of derivability. It was Cauchy who, through Cauchy's formula, showed that a derivable application of the complex variable is in fact indefinitely derivable, and even analytic.

One of the consequences of this fundamental observation is the great rigidity of the very notion of derivable application of the complex variable (also called holomorphic application). Another, more fortunate one, is that it allows many calculations of integrals using the residue formula, one of the avatars of Cauchy's formula.

Complex analysis of one variable found new developments in the study of harmonic applications of two real variables and, a little later, in its applications to arithmetic. The study of the zeta function and, in particular, the location of the points where it cancels out, leads to the prime number theorem, which specifies its distribution. Riemann's conjecture, still valid today, is in the same vein. On the other hand, staying in the more down-to-earth realm of physics applications, many special functions, defined in a part...