First of all, let me give some definitions, which can be understood by every student who have studied elementary mathematical analysis and complex analysis.
Definition 1
Let
be an ascending sequence. the upper density of 
is
The step of
is
Definition 2
The generalized Dirichlet series
, with 
is given by
Lemma 1
If
does not converge or diverge for all s, there exists 
such that 
converges absolutely if 
and it does not converge absolutely if 
.
is the abscissa of absolute convergence of 
.Theorem due to S. Mendelbrojt
Let
be a generalized dirichlet generating function with abscissa of absolute convergence being finite. For 
and 
there exists a continuous function 
with 
such that for all 
, 
has a singular point in the rectangle 
. One such function is
for 
.Corollary
In particular, if
and 
and is finite, then every point on the abscissa of absolute convergence is a singular point. So no analytic continuation of the series from right halfplane to left halfplane is possible.
No comments:
Post a Comment