An approach for solving a variety of coefficient inverse problems for the Sturm-Liouville equation \(-y^{\prime \prime }+q(x)y=\rho ^{2}y\) with a complex valued potential \(q(x)\) is presented. It is based on analytic series representations for solutions, obtained by expanding the integral kernels of corresponding transmutation operators into series of orthogonal polynomials. For problems on finite intervals, the Neumann series of Bessel functions (NSBF) representations for solutions are involved. 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 first NSBF 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. For problems on infinite intervals the power series expansions of the Jost solutions are involved, with respect to a parameter z, related to the spectral parameter by a Möbius map. The overall approach leads to efficient numerical algorithms for solving a wide variety of coefficient inverse problems.