  In this work we provide a dispersive analysis of $\pi\pi \rightarrow
K\bar{K}$ scattering. For this purpose we present a set of partial-wave
hyperbolic dispersion relations using a family of hyperbolas that maximizes the
applicability range of the hyperbolic dispersive representation, which we have
extended up to 1.47 GeV. We then use these equations first to test simple fits
to different and often conflicting data sets, also showing that some of these
data and some popular parameterizations of these waves fail to satisfy the
dispersive analysis. Our main result is obtained after imposing these new
relations as constraints on the data fits. We thus provide simple and precise
parameterizations for the S, P and D waves that describe the experimental data
from $K\bar K$ threshold up to 2 GeV, while being consistent with crossing
symmetric partial-wave dispersion relations up to their maximum applicability
range of 1.47 GeV. For the $S$-wave we have found that two solutions describing
two conflicting data sets are possible. The dispersion relations also provide a
representation for $S$, $P$ and $D$ waves in the pseudo-physical region.
