Consider a power series in a complex variable 
 |
(1)
|
that is convergent within the open disk 
. Convergence is limited to within 
by the presence of at least one singularity
on the boundary 
of 
. If the singularities on 
are so densely packed that analytic
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 the functional 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 a lacunary function.
