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Principle of Permanence


In its simplest form, the principle of permanence states that, given any analytic function f(z) defined on an open (and connected) set U of the complex numbers C, and a convergent sequence {a_n} which along with its limit L belongs to U, such that f(a_n)=0 for all n, then f(z) is uniformly zero on U.

This is easily proved by showing that the Taylor series of f(z) about L must have all its coefficients equal to 0.

The principle of permanence has wide-ranging consequences. For example, if G and H are analytic functions defined on U, then any functional equation of the form

 G(f(z))=H(f(z))

that is true for all z in a closed subset of U having a limit point in U (e.g., a nonempty open subset of U) must be true for all z in U.


See also

Analytic Function

This entry contributed by Daniel Asimov

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References

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

Referenced on Wolfram|Alpha

Principle of Permanence

Cite this as:

Asimov, Daniel. "Principle of Permanence." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PrincipleofPermanence.html

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