persistence of analytic relations
The principle of persistence of analytic relations states that any algebraic relation between several analytic functions which holds on a sufficiently large set also holds wherever the functions are defined.
A more explicit statement of this principle is as follows: Let be complex analytic functions. Suppose that there exists an open set on which all these functions are defined and that there exists a polynomial of variables such that whenever lies in a subset of which has a limit point in . Then for all .
This fact is a simple consequence of the rigidity theorem for analytic functions. If are all analytic in , then is also analytic in . Hence, if when in , then for all .
This principle is very useful in establishing identites involving analytic functions because it means that it suffices to show that the identity holds on a small subset. For instance, from the fact that the familiar identity holds for all real , it automatically holds for all complex values of . This principle also means that it is unnecessary to specify for which values of the variable an algebraic relation between analytic functions holds since, if such a relation holds, it will hold for all values for which the functions appearing in the relation are defined.
|Title||persistence of analytic relations|
|Date of creation||2013-03-22 14:44:17|
|Last modified on||2013-03-22 14:44:17|
|Last modified by||rspuzio (6075)|