persistence of analytic relations

The principle of persistence of analytic relations states that any algebraic relation between several analytic functionsMathworldPlanetmath 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 f1,f2,fn be complex analytic functions. Suppose that there exists an open set D on which all these functions are defined and that there exists a polynomial p of n variables such that p(f1(z),f2(z),,fn(z))=0 whenever z lies in a subset X of D which has a limit pointMathworldPlanetmathPlanetmath in D. Then p(f1(z),f2(z),fn(z))=0 for all zD.

This fact is a simple consequence of the rigidity theorem for analytic functions. If f1,f2,fn are all analytic in D, then p(f1(z),f2(z),fn(z)) is also analytic in D. Hence, if p(f1(z),f2(z),,fn(z))=0 when z in X, then p(f1(z),f2(z),,fn(z))=0 for all zD.

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 sin2x+cos2x=1 holds for all real x, it automatically holds for all complex values of x. 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
Canonical name PersistenceOfAnalyticRelations
Date of creation 2013-03-22 14:44:17
Last modified on 2013-03-22 14:44:17
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Theorem
Classification msc 30A99
Related topic ComplexSineAndCosine