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 $f_{1},f_{2},\ldots f_{n}$ 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 $p(f_{1}(z),f_{2}(z),\ldots,f_{n}(z))=0$ whenever $z$ lies in a subset $X$ of $D$ which has a limit point in $D$ . Then $p(f_{1}(z),f_{2}(z),\ldots f_{n}(z))=0$ for all $z\in D$ .

This fact is a simple consequence of the rigidity theorem for analytic functions. If $f_{1},f_{2},\ldots f_{n}$ are all analytic in $D$ , then $p(f_{1}(z),f_{2}(z),\ldots f_{n}(z))$ is also analytic in $D$ . Hence, if $p(f_{1}(z),f_{2}(z),\ldots,f_{n}(z))=0$ when $z$ in $X$ , then $p(f_{1}(z),f_{2}(z),\ldots,f_{n}(z))=0$ for all $z\in D$ .

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 $\sin^{2}x+\cos^{2}x=1$ 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