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Natural Boundary
Consider apower series in a complex variable
 | (1) |
that is convergent within theopen disk
. Convergence is limited to within
by the presence of at least onesingularity on theboundary
of
. If the singularities on
are so densely packed thatanalytic continuation cannot be carried out on a path that crosses
, then
is said to form a natural boundary (or "natural boundary of analyticity") for the function
.
As an example, consider the function
 | (2) |
Then
formally satisfies thefunctional equation
 | (3) |
The series (◇) clearly converges within
. Now consider
. Equation (◇) tells us that
which can only be satisfied if
. Considering now
, equation (◇) becomes
and hence
. Substituting
for
in equation (◇) then gives
 | (4) |
from which it follows that
 | (5) |
Now consider
equal to any of the fourth roots of unity,
,
, for example
. Then
. Applying this procedure recursively shows that
is infinite for any
such that
with
, 1, 2, .... In any arc of the circle
of finite length there will therefore be an infinite number of points for which
is infinite and so
constitutes a natural boundary for
.
A function that has a natural boundary is said to be alacunaryfunction.
See also
Analytic Continuation,Analytic Function,Boundary,Branch Cut,Lacunary Function,Natural Domain,SingularityThis entry contributed byJonathanDeane
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References
Ash, R. B. Ch. 3 inComplex Variables. New York: Academic Press, 1971.Knopp, K.Theory of Functions, Parts I and II. New York: Dover, Part I, p. 101, 1996.Referenced on Wolfram|Alpha
Natural BoundaryCite this as:
Deane, Jonathan. "Natural Boundary." FromMathWorld--A Wolfram Resource, created byEric W. Weisstein.https://mathworld.wolfram.com/NaturalBoundary.html