where
and .
Employing the partial–fraction expansion of the meromorphic function
[56,
Problem 5.2.1], we obtain
(2.54)
where
and
().
Taking into account the identity [56, Problem 5.2.4]
and eliminating
in (2.54), and after some transformations, we obtain the final formula for
, valid
for all :
(2.55)
The series in (2.55) converges in square .
From the divergence of the harmonic series and the estimate
it follows that the series in (2.55) is divergent at
. Hence, the kernel
is unbounded:
. The plots of the
kernel for two
admissible values of
are depicted in Figures 2.5 and 2.6.
Figure 2.5:
Plot of the kernel
for
Figure 2.6:
Plot of the kernel
for
Numerical calculations show that the folding of
increases for
decreasing
from
to . For
the
kernel
may take negative values.
Remark 2.8.1. Using (2.25) and (2.43), it is possible to characterize the kernel
by the
integral equation
here, denotes
the kernel of
(see (2.31)), and
is defined by (2.8). An approximate solution of this equation can be found with the aid of the
inflation technique [46] which allows us to pass to the matrix Lyapunov equation.