Analytical Prolongation

In Analyze complexes, the theory of the analytical prolongation details the whole of the properties and techniques concerning the prolongation of the functions Holomorphe S (or analytical). She considers initially the question of the prolongation in the complex plan. Then it approaches more general forms of extension which make it possible to take into account the Singularité S and the complications topological S which accompanies them. The theory then utilizes either the rather old and not very operative concept of multiform function , or the more powerful concept of Surface of Riemann.

There exists also a theory of the analytical prolongation for the functions with several complex variables, whose difficulty is larger, and whose treatment was at the origin of the introduction of the Cohomologie of the beams.

Holomorphic function on open of the complex plan

Position of the problems of analytical prolongation

Being given an analytical function complexes in a field D, the theory puts primarily two questions: on the one hand which is the greatest field where the representation of the function is valid (ex: if the function is defined by a whole series, the ray of convergence of this series; if the function is defined by an integral or a differential equation,… the field of validity of this representation.) then if the representation can be wide with a wider field, even at the price of an extension of the representation (related concepts: integral taken within the meaning of the principal parts of Cauchy, pseudo functions of Hadamard, radial prolongation, star of Mittag-Lefler, summation of the divergent series within the meaning of Césaro, Borel,…).

Unicity of the analytical prolongation

One lays out of this result on the analytical functions. Are $U \ subset \ mathbb {C}$ open, $a$ a point of $U$ and an analytical function $f: U \ to \ mathbb {C}$ . It is supposed moreover that $U$ is related (this assumption is essential). Then the three following proposals are equivalent:

$f$ is identically null on $U$
$f$ is identically null in a vicinity of $a$
$\ forall N \ in \ NR, \ f^ {(N)}(a)=0$

This theorem then means that if analytical function on open related is cancelled on a disc of so small ray it is, it is the null function.

By applying this theorem to the difference in two analytical functions, one obtains the unicity of the analytical prolongation. Indeed, if $f, g$ are two analytical functions on open a $U$ related of $\ mathbb {C}$ and if $f$ and $g$ coincide on a point neighborhood of $U$ , then $f-g=0$ on this vicinity thus by theorem $f-g=0$ on $U$ and thus $f=g$ on $U$ .

Intervention of the singularities

That is to say F an analytical function on open a U . It is natural to seek to prolong F at the points of the border of U . That is to say U such a point.

For typology, it is important to separate in the questions from existence and unicity the local and total points of view. For example one can define a function Logarithme complexes holomorphic on the private level of a formed half-line of negative realities, and no holomorphic extension to a larger field exists. However, if one considers a strictly negative reality given U , and the restriction of the function on the complexes of strictly positive imaginary part, this restriction, it, can be prolonged on a disc centered on the point U , and this prolongation is the only possible one on the field considered (meeting of a half-plane and a disc).

More generally if, even if it means to make such a preliminary restriction, there exists a holomorphic function in the vicinity of U who prolongs F , the point U will be known as regular .

Definition: U is regular for F when there exists an open unit related V container U and a holomorphic application G on V , such as F and G coincides on open a W included in the two fields of definition and having U for border. In the contrary case, the point is known as singular .

The example of the complex logarithm shows that the concept of regular point is not interpreted like a prolongation of the initial function, but only like one local possibility to prolong it. The topology of open the U intervenes to determine if the total prolongation is indeed possible.

Multiform functions

Surface of Riemann associated with a function

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