  An approach for solving a variety of inverse coefficient problems for the
Sturm-Liouville equation -y''+q(x)y={\lambda}y with a complex valued potential
q(x) is presented. It is based on Neumann series of Bessel functions
representations for solutions. With their aid the problem is reduced to a
system of linear algebraic equations for the coefficients of the
representations. The potential is recovered from an arithmetic combination of
the first two coefficients. Special cases of the considered problems include
the recovery of the potential from a Weyl function, inverse two-spectra
Sturm-Liouville problems, as well as the inverse scattering problem on a finite
interval. The approach leads to efficient numerical algorithms for solving
coefficient inverse problems. Numerical efficiency is illustrated by several
examples.
