  A spectral parameter power series (SPPS) representation for regular solutions
of singular Bessel type Sturm-Liouville equations with complex coefficients is
obtained as well as an SPPS representation for the (entire) characteristic
function of the corresponding spectral problem on a finite interval. It is
proved that the set of zeros of the characteristic function coincides with the
set of all eigenvalues of the Sturm-Liouville problem. Based on the SPPS
representation a new mapping property of the transmutation operator for the
considered perturbed Bessel operator is obtained, and a new numerical method
for solving corresponding spectral problems is developed. The range of
applicability of the method includes complex coefficients, complex spectrum and
equations in which the spectral parameter stands at a first order linear
differential operator. On a set of known test problems we show that the
developed numerical method based on the SPPS representation is highly
competitive in comparison to the best available solvers such as SLEIGN2,
MATSLISE and some other codes and give an example of an exactly solvable test
problem admitting complex eigenvalues to which the mentioned solvers are not
applicable meanwhile the SPPS method delivers excellent numerical results.
