Fabry theorem

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 [Fa].

References

[Bi]	L. Bieberbach, "Analytische Fortsetzung" , Springer (1955)
[Di]	P. Dienes, "The Taylor series" , Oxford Univ. Press & Dover (1957)
[Fa]	E. Fabry, "Sur les points singuliers d'une fonction donnée 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 Zbl 27.0303.01
[La]	E. Landau, "Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)
[Le]	A.F. Leont'ev, "Exponential series" , Moscow (1976) (In Russian)