  We present a new approach to quantifying pole parameters of single-channel
processes based on Laurent expansion of partial wave T-matrices. Instead of
guessing the analytical form of non-singular part of Laurent expansion as it is
usually done, we represent it by the convergent series of Pietarinen functions.
As the analytic structure of non-singular term is usually very well known
(physical cuts with branhcpoints at inelastic thresholds, and unphysical cuts
in the negative energy plane), we show that we need one Pietarinen series per
cut, and the number of terms in each Pietarinen series is automatically
determined by the quality of the fit. The method is tested on a toy model
constructed from two known poles, various background terms, and two physical
cuts, and shown to be robust and confident up to three Pietarinen series. We
also apply this method to Zagreb CMB amplitudes for the N(1535) 1/2- resonance,
and confirm the full success of the method on realistic data. This formalism
can also be used for fitting experimental data, and the procedure is very
similar as when Breit-Wigner functions are used, but with one modification:
Laurent expansion with Pietarinen series is replacing the standard Breit-Wigner
T-matrix form.
