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 $f_1,f_2,\mathrm{\dots }{f}_n$ 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 $p(f_1(z),f_2(z),\mathrm{\dots },f_n(z))=0$ whenever $z$ lies in a subset $X$ of $D$ which has a limit point in $D$. Then $p(f_1(z),f_2(z),\mathrm{\dots }{f}_n(z))=0$ for all $z\in D$.

This fact is a simple consequence of the rigidity theorem for analytic functions. If $f_1,f_2,\mathrm{\dots }{f}_n$ are all analytic in $D$, then $p(f_1(z),f_2(z),\mathrm{\dots }{f}_n(z))$ is also analytic in $D$. Hence, if $p(f_1(z),f_2(z),\mathrm{\dots },f_n(z))=0$ when $z$ in $X$, then $p(f_1(z),f_2(z),\mathrm{\dots },f_n(z))=0$ 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.