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.