In
mathematics, the Fabry gap theorem is a result about the
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
anywhere on the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
of its
disc of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
.
The theorem may be deduced from the first main theorem of
Turán's method.
Statement of the theorem
Let 0 < ''p''
1 < ''p''
2 < ... be a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s such that the sequence ''p''
''n''/''n'' diverges to ∞. Let (''α''
''j'')
''j''∈N be a sequence of complex numbers such that the power series
:
has radius of convergence 1. Then the unit circle is a
natural boundary for the series ''f''.
Converse
A converse to the theorem was established by
George Pólya
George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamenta ...
. If lim inf ''p''
''n''/''n'' is finite then there exists a power series with exponent sequence ''p''
''n'', radius of convergence equal to 1, but for which the unit circle is not a natural boundary.
See also
*
Gap theorem (disambiguation)
*
Lacunary function In analysis, a lacunary function, also known as a lacunary series, is an analytic function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined by a power series. The word ''lacunary'' is deriv ...
*
Ostrowski–Hadamard gap theorem In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense ...
References
*
* {{cite journal , last=Erdős , first=Pál , authorlink=Paul Erdős , title=Note on the converse of Fabry's gap theorem , journal=
Transactions of the American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 ...
, volume=57 , pages=102–104 , year=1945 , issn=0002-9947 , jstor=1990169 , zbl=0060.20303 , doi=10.2307/1990169
Mathematical series
Theorems in complex analysis