View source for Fabry theorem

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==Fabry's gap theorem==

If the exponents $\lambda_n$ in the power series

$$ f(z)=\sum_{n=1}^\infty a_nz^{\lambda_n},$$

with radius of convergence $R$, $0<R<\infty$, satisfy the condition

$$\lim_{n\to\infty}\frac{n}{\lambda_n}=0,$$

then the circle $\lvert z\rvert=R$ is a [[natural boundary]]: all points of the cicle are singular points for $f(z)$. The theorem can be generalized to Dirichlet series.

A converse to the theorem was established by George Pólya.  If $\lim\inf \lambda_n/n$ 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.

==Fabry's quotient theorem==

If the coefficients in the power series

$$ f(z)=\sum_{n=0}^\infty a_nz^n,$$

with unit radius of convergence, satisfy the condition

$$ \lim_{n\to \infty}\frac{a_n}{a_{n+1}}=s,$$

then $z=s$ is a singular point of $f(z)$.

These theorems were obtained by E. Fabry {{Cite|Fa}}.

====References====
{|
|-
|valign="top"|{{Ref|Bi}}|| valign="top"| L. Bieberbach,   "Analytische Fortsetzung" , Springer  (1955)
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|valign="top"|{{Ref|Di}}|| valign="top"|  P. Dienes,   "The Taylor series" , Oxford Univ. Press &amp; Dover  (1957)
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|valign="top"|{{Ref|Fa}}|| valign="top"| E. Fabry,   "Sur les points singuliers d'une fonction donée par son développement en série et l'impossibilité du prolongement analytique dans des cas très généraux"  ''Ann. Sci. Ecole Norm. Sup.'' , '''13'''  (1896)  pp. 367–399
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|valign="top"|{{Ref|La}}|| valign="top"|  E. Landau,   "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie" , ''Das Kontinuum und andere Monographien'' , Chelsea, reprint  (1973)
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|valign="top"|{{Ref|Le}}|| valign="top"|  A.F. Leont'ev,   "Exponential series" , Moscow  (1976)  (In Russian)
|-
|}