Principle of Permanence
المؤلف:
Wolfram, S
المصدر:
A New Kind of Science. Champaign, IL: Wolfram Media
الجزء والصفحة:
p. 1168
11-12-2018
975
Principle of Permanence
In its simplest form, the principle of permanence states that, given any analytic function 
defined on an open (and connected) set 
of the complex numbers 
, and a convergent sequence ![<span style=]()
{a_n}" src="http://mathworld.wolfram.com/images/equations/PrincipleofPermanence/Inline4.gif" style="height:14px; width:23px" /> which along with its limit 
belongs to 
, such that 
for all 
, then 
is uniformly zero on 
.
This is easily proved by showing that the Taylor series of 
about 
must have all its coefficients equal to 0.
The principle of permanence has wide-ranging consequences. For example, if 
and 
are analytic functions defined on 
, then any functional equation of the form
that is true for all 
in a closed subset of 
having a limit point in 
(e.g., a nonempty open subset of 
) must be true for all 
in 
.
REFERENCES:
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.
الاكثر قراءة في التحليل العقدي
اخر الاخبار
اخبار العتبة العباسية المقدسة
