The principle of permanence in complex analysis is the persistence of identities between analytic
functions, as expressed by the identity theorem (Boisgérault 2017, pp. 91-92).
If an analytic function 
is defined on a connected open set 
and vanishes at every point of a convergent
sequence 
of distinct points in 
whose limit 
also belongs to 
, then 
is identically zero on
.
This follows from the fact that an analytic function which is not identically zero has isolated zeros. In terms of the Taylor
series about 
,
a first nonzero coefficient would make 
an isolated zero, contradicting the
distinct zeros 
approaching 
.
For example, let 
be an analytic function
on 
,
and let 
and 
be analytic functions on an open
set 
containing 
.
The functional equation
 |
then holds throughout 
whenever it holds on a subset of
with a limit point in 
, such as a nonempty open subset
of 
.
Indeed, 
is an analytic function on 
, so the identity theorem applies to this difference.
The same name also refers to a different algebraic guideline, associated with Peacock, for preserving identities when extending number systems (Wolfram 2002, p. 1168). This algebraic usage is distinct from the complex analysis theorem above.
