Complex analysis - Holomorphic applications

Holomorphic applications can be used to elucidate certain phenomena which, at first glance, seem to involve only real numbers, whereas these applications are defined on an open of the complex plane.

A striking example of this is the calculation of integrals of functions of the real variable, made simple and above all systematic by the use of the residue formula.

In computational terms, this formula expresses the geometry of the complex plane, which differs from that of the real straight line in that, in the former, a point can be surrounded by a yaw (i.e., a curve that closes in on itself). The notion of integral along a yaw then enables us to calculate an integral "around" a pole of a holomorphic application f. In doing so, two terms emerge:

• the first, of a geometrical nature, is the number of turns the yaw makes around the pole: this is the notion of index ;

• the second expresses the behavior of f near the pole, which involves a number, the residual of f at this pole.

Using such integrals, we obtain a fairly general formula, known as the residue formula. Suitably applied to particular laces, it can be used to obtain the value of many integrals of applications defined on $ℝ$ , often restricting certain holomorphic applications on $ℂ$ .

We can also deduce other equalities by applying the residue formula to applications dependent on a complex parameter. These equalities give rise to identities between complex functions (of the parameter). Eulerian developments are of this nature.

The use of integrals along certain paths also leads to the solution of differential equations. This subject, in itself very vast, is not dealt with in the article, nor is the search for the asymptotic behavior of integrals depending on a parameter.

Holomorphic application theory" is covered in the fascicule