  Studies on the IJ=00 $\pi\pi$ and $K\bar K$ coupled--channel system are made
using newly derived dispersion relations between the phase shifts and poles and
cuts. It is found that the $\sigma$ resonance must be introduced to explain the
experimental phase shifts, after evaluating the cut contribution. The effects
of nearby branch point singularities to the determination of the $f_0(980)$
resonance are also carefully clarified.
