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Natural Boundary


Consider a power series in a complex variable z

 g(z)=sum_(n=0)^inftya_nz^n
(1)

that is convergent within the open disk D:|z|<R. Convergence is limited to within D by the presence of at least one singularity on the boundary partialD of D. If the singularities on D are so densely packed that analytic continuation cannot be carried out on a path that crosses D, then D is said to form a natural boundary (or "natural boundary of analyticity") for the function g(z).

As an example, consider the function

 f(z)=sum_(n=0)^inftyz^(2^n)=z+z^2+z^4+....
(2)

Then f(z) formally satisfies the functional equation

 f(z)=z+f(z^2).
(3)

The series (◇) clearly converges within D_1:|z|<1. Now consider z=1. Equation (◇) tells us that f(1)=1+f(1) which can only be satisfied if f(1)=infty. Considering now z=-1, equation (◇) becomes f(-1)=-1+infty and hence f(-1)=infty. Substituting z^2 for z in equation (◇) then gives

 f(z^2)=z^2+f(z^4)=f(z)-z.
(4)

from which it follows that

 f(z)=z+z^2+f(z^4).
(5)

Now consider z equal to any of the fourth roots of unity, +/-1, +/-i, for example z=-i. Then f(-i)=-i-1+f(1)=infty. Applying this procedure recursively shows that f(z) is infinite for any z such that z^(2^n)=1 with n=0, 1, 2, .... In any arc of the circle partialD_1 of finite length there will therefore be an infinite number of points for which f(z) is infinite and so D_1 constitutes a natural boundary for f(z).

A function that has a natural boundary is said to be a lacunary function.


See also

Analytic Continuation, Analytic Function, Boundary, Branch Cut, Lacunary Function, Natural Domain, Singularity

This entry contributed by Jonathan Deane

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References

Ash, R. B. Ch. 3 in Complex Variables. New York: Academic Press, 1971.Knopp, K. Theory of Functions, Parts I and II. New York: Dover, Part I, p. 101, 1996.

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Natural Boundary

Cite this as:

Deane, Jonathan. "Natural Boundary." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NaturalBoundary.html

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